/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 176 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 16.4 s] (14) BOUNDS(1, INF) (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) NarrowingProof [BOTH BOUNDS(ID, ID), 41 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 9 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 20 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 20 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 104 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 33 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 36 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 19 ms] (148) CdtProblem (149) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 4 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 36 ms] (168) CdtProblem (169) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(not, app(not, x)) -> x app(not, app(app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) app(not, app(app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) app(app(and, x), app(app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(and, app(app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(not, app(not, x)) -> x app(not, app(app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) app(not, app(app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) app(app(and, x), app(app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(and, app(app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The (relative) TRS S consists of the following rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(not, app(not, x)) -> x app(not, app(app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) app(not, app(app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) app(app(and, x), app(app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(and, app(app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The (relative) TRS S consists of the following rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(not, app(not, x)) -> x app(not, app(app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) app(not, app(app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) app(app(and, x), app(app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(and, app(app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The (relative) TRS S consists of the following rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(not, app(not, x)) -> x app(not, app(app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) app(not, app(app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) app(app(and, x), app(app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(and, app(app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The (relative) TRS S consists of the following rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: app(not, app(not, x)) -> x app(not, app(app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) app(not, app(app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) app(app(and, x), app(app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(and, app(app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app not :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app or :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app and :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app map :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app nil :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app cons :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app filter :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app true :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app false :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encArg :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_not :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_or :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_and :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app hole_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app1_0 :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0 :: Nat -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (12) Obligation: Innermost TRS: Rules: app(not, app(not, x)) -> x app(not, app(app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) app(not, app(app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) app(app(and, x), app(app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(and, app(app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app not :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app or :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app and :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app map :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app nil :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app cons :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app filter :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app true :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app false :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encArg :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_not :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_or :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_and :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app hole_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app1_0 :: not:or:and:map:nil:cons:filter:filter2:true:false:cons_app gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0 :: Nat -> not:or:and:map:nil:cons:filter:filter2:true:false:cons_app Generator Equations: gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0(0) <=> not gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0(+(x, 1)) <=> cons_app(not, gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0(x)) The following defined symbols remain to be analysed: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0(n18591_0)) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0(0)) Induction Step: encArg(gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0(+(n18591_0, 1))) ->_R^Omega(0) app(encArg(not), encArg(gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0(n18591_0))) ->_R^Omega(0) app(not, encArg(gen_not:or:and:map:nil:cons:filter:filter2:true:false:cons_app2_0(n18591_0))) ->_IH app(not, *3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (14) BOUNDS(1, INF) ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(not, app(not, x)) -> x app(not, app(app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) app(not, app(app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) app(app(and, x), app(app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(and, app(app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(c_app(and, c_app(c_app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(not, c_app(not, x)) -> x app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs) app(c_app(map, f), nil) -> nil app(c_app(filter, f), nil) -> nil app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(not, c_app(c_app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) app(not, c_app(c_app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) app(c_app(and, x), c_app(c_app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) The (relative) TRS S consists of the following rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(x0, x1) -> c_app(x0, x1) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(c_app(and, c_app(c_app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) [1] app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(not, c_app(not, x)) -> x [1] app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) [1] app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs) [1] app(c_app(map, f), nil) -> nil [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) [1] app(not, c_app(c_app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) [1] app(not, c_app(c_app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) [1] app(c_app(and, x), c_app(c_app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) [1] encArg(not) -> not [0] encArg(or) -> or [0] encArg(and) -> and [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_not -> not [0] encode_or -> or [0] encode_and -> and [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(and, c_app(c_app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) [1] app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(not, c_app(not, x)) -> x [1] app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) [1] app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs) [1] app(c_app(map, f), nil) -> nil [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) [1] app(not, c_app(c_app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) [1] app(not, c_app(c_app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) [1] app(c_app(and, x), c_app(c_app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) [1] encArg(not) -> not [0] encArg(or) -> or [0] encArg(and) -> and [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_not -> not [0] encode_or -> or [0] encode_and -> and [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] The TRS has the following type information: app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app c_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encArg :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app cons_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app encode_false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app Rewrite Strategy: INNERMOST ---------------------------------------- (23) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: encArg_1 encode_app_2 encode_not encode_or encode_and encode_map encode_nil encode_cons encode_filter encode_filter2 encode_true encode_false app_2 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_not -> null_encode_not [0] encode_or -> null_encode_or [0] encode_and -> null_encode_and [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_not, null_encode_or, null_encode_and, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(and, c_app(c_app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) [1] app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(not, c_app(not, x)) -> x [1] app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) [1] app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs) [1] app(c_app(map, f), nil) -> nil [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) [1] app(not, c_app(c_app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) [1] app(not, c_app(c_app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) [1] app(c_app(and, x), c_app(c_app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) [1] encArg(not) -> not [0] encArg(or) -> or [0] encArg(and) -> and [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_not -> not [0] encode_or -> or [0] encode_and -> and [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_not -> null_encode_not [0] encode_or -> null_encode_or [0] encode_and -> null_encode_and [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app c_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encArg :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (25) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (26) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(and, c_app(c_app(or, y), z)), x) -> app(app(or, app(c_app(and, x), y)), app(c_app(and, x), z)) [1] app(c_app(and, c_app(c_app(or, y), z)), x) -> app(app(or, app(c_app(and, x), y)), app(null_app, z)) [1] app(c_app(and, c_app(c_app(or, y), z)), x) -> app(app(or, app(null_app, y)), app(c_app(and, x), z)) [1] app(c_app(and, c_app(c_app(or, y), z)), x) -> app(app(or, app(null_app, y)), app(null_app, z)) [1] app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(not, c_app(not, x)) -> x [1] app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(c_app(cons, x), app(c_app(filter, f), xs)) [1] app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(c_app(cons, x), app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(null_app, app(c_app(filter, f), xs)) [1] app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(null_app, app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(c_app(filter, f), xs) [1] app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(null_app, xs) [1] app(c_app(map, f), nil) -> nil [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(filter, c_app(and, c_app(c_app(or, y3), z1))), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(or, app(app(and, x), y3)), app(app(and, x), z1))), c_app(and, c_app(c_app(or, y3), z1))), x), xs) [2] app(c_app(filter, c_app(map, f5)), c_app(c_app(cons, c_app(c_app(cons, x8), xs1)), xs)) -> app(app(app(app(filter2, app(app(cons, app(f5, x8)), app(app(map, f5), xs1))), c_app(map, f5)), c_app(c_app(cons, x8), xs1)), xs) [2] app(c_app(filter, not), c_app(c_app(cons, c_app(not, x9)), xs)) -> app(app(app(app(filter2, x9), not), c_app(not, x9)), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, true), f6), x10)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(cons, x10), app(app(filter, f6), x))), c_app(c_app(c_app(filter2, true), f6), x10)), x), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, false), f7), x11)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(filter, f7), x)), c_app(c_app(c_app(filter2, false), f7), x11)), x), xs) [2] app(c_app(filter, c_app(map, f8)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(map, f8)), nil), xs) [2] app(c_app(filter, c_app(filter, f9)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(filter, f9)), nil), xs) [2] app(c_app(filter, c_app(filter, f10)), c_app(c_app(cons, c_app(c_app(cons, x12), xs2)), xs)) -> app(app(app(app(filter2, app(app(app(app(filter2, app(f10, x12)), f10), x12), xs2)), c_app(filter, f10)), c_app(c_app(cons, x12), xs2)), xs) [2] app(c_app(filter, not), c_app(c_app(cons, c_app(c_app(or, x13), y4)), xs)) -> app(app(app(app(filter2, app(app(and, app(not, x13)), app(not, y4))), not), c_app(c_app(or, x13), y4)), xs) [2] app(c_app(filter, not), c_app(c_app(cons, c_app(c_app(and, x14), y5)), xs)) -> app(app(app(app(filter2, app(app(or, app(not, x14)), app(not, y5))), not), c_app(c_app(and, x14), y5)), xs) [2] app(c_app(filter, c_app(and, x15)), c_app(c_app(cons, c_app(c_app(or, y6), z2)), xs)) -> app(app(app(app(filter2, app(app(or, app(app(and, x15), y6)), app(app(and, x15), z2))), c_app(and, x15)), c_app(c_app(or, y6), z2)), xs) [2] app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, c_app(f, x)), f), x), xs) [1] app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, null_app), f), x), xs) [1] app(not, c_app(c_app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) [1] app(not, c_app(c_app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) [1] app(c_app(and, x), c_app(c_app(or, y), z)) -> app(app(or, app(c_app(and, x), y)), app(c_app(and, x), z)) [1] app(c_app(and, x), c_app(c_app(or, y), z)) -> app(app(or, app(c_app(and, x), y)), app(null_app, z)) [1] app(c_app(and, x), c_app(c_app(or, y), z)) -> app(app(or, app(null_app, y)), app(c_app(and, x), z)) [1] app(c_app(and, x), c_app(c_app(or, y), z)) -> app(app(or, app(null_app, y)), app(null_app, z)) [1] encArg(not) -> not [0] encArg(or) -> or [0] encArg(and) -> and [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_not -> not [0] encode_or -> or [0] encode_and -> and [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_not -> null_encode_not [0] encode_or -> null_encode_or [0] encode_and -> null_encode_and [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app c_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encArg :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: and => 0 or => 8 map => 5 cons => 1 not => 7 filter2 => 4 true => 9 filter => 3 false => 2 nil => 6 null_encArg => 0 null_encode_app => 0 null_encode_not => 0 null_encode_or => 0 null_encode_and => 0 null_encode_map => 0 null_encode_nil => 0 null_encode_cons => 0 null_encode_filter => 0 null_encode_filter2 => 0 null_encode_true => 0 null_encode_false => 0 null_app => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> x :|: z'' = 1 + 7 + x, x >= 0, z' = 7 app(z', z'') -{ 2 }-> app(app(app(app(4, x9), 7), 1 + 7 + x9), xs) :|: xs >= 0, z' = 1 + 3 + 7, z'' = 1 + (1 + 1 + (1 + 7 + x9)) + xs, x9 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(app(app(4, app(f10, x12)), f10), x12), xs2)), 1 + 3 + f10), 1 + (1 + 1 + x12) + xs2), xs) :|: xs >= 0, xs2 >= 0, f10 >= 0, x12 >= 0, z' = 1 + 3 + (1 + 3 + f10), z'' = 1 + (1 + 1 + (1 + (1 + 1 + x12) + xs2)) + xs app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(8, app(app(0, x), y3)), app(app(0, x), z1))), 1 + 0 + (1 + (1 + 8 + y3) + z1)), x), xs) :|: xs >= 0, z1 >= 0, x >= 0, y3 >= 0, z' = 1 + 3 + (1 + 0 + (1 + (1 + 8 + y3) + z1)), z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(8, app(app(0, x15), y6)), app(app(0, x15), z2))), 1 + 0 + x15), 1 + (1 + 8 + y6) + z2), xs) :|: z'' = 1 + (1 + 1 + (1 + (1 + 8 + y6) + z2)) + xs, xs >= 0, y6 >= 0, z' = 1 + 3 + (1 + 0 + x15), x15 >= 0, z2 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(8, app(7, x14)), app(7, y5))), 7), 1 + (1 + 0 + x14) + y5), xs) :|: y5 >= 0, xs >= 0, z' = 1 + 3 + 7, z'' = 1 + (1 + 1 + (1 + (1 + 0 + x14) + y5)) + xs, x14 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(3, f7), x)), 1 + (1 + (1 + 4 + 2) + f7) + x11), x), xs) :|: xs >= 0, z' = 1 + 3 + (1 + (1 + (1 + 4 + 2) + f7) + x11), x >= 0, f7 >= 0, x11 >= 0, z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(1, x10), app(app(3, f6), x))), 1 + (1 + (1 + 4 + 9) + f6) + x10), x), xs) :|: z' = 1 + 3 + (1 + (1 + (1 + 4 + 9) + f6) + x10), xs >= 0, x >= 0, f6 >= 0, x10 >= 0, z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(1, app(f5, x8)), app(app(5, f5), xs1))), 1 + 5 + f5), 1 + (1 + 1 + x8) + xs1), xs) :|: xs >= 0, z'' = 1 + (1 + 1 + (1 + (1 + 1 + x8) + xs1)) + xs, x8 >= 0, z' = 1 + 3 + (1 + 5 + f5), xs1 >= 0, f5 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(0, app(7, x13)), app(7, y4))), 7), 1 + (1 + 8 + x13) + y4), xs) :|: x13 >= 0, xs >= 0, z' = 1 + 3 + 7, z'' = 1 + (1 + 1 + (1 + (1 + 8 + x13) + y4)) + xs, y4 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, 6), 1 + 5 + f8), 6), xs) :|: f8 >= 0, xs >= 0, z'' = 1 + (1 + 1 + 6) + xs, z' = 1 + 3 + (1 + 5 + f8) app(z', z'') -{ 2 }-> app(app(app(app(4, 6), 1 + 3 + f9), 6), xs) :|: xs >= 0, z'' = 1 + (1 + 1 + 6) + xs, f9 >= 0, z' = 1 + 3 + (1 + 3 + f9) app(z', z'') -{ 1 }-> app(app(app(app(4, 0), f), x), xs) :|: f >= 0, xs >= 0, x >= 0, z'' = 1 + (1 + 1 + x) + xs, z' = 1 + 3 + f app(z', z'') -{ 1 }-> app(app(app(app(4, 1 + f + x), f), x), xs) :|: f >= 0, xs >= 0, x >= 0, z'' = 1 + (1 + 1 + x) + xs, z' = 1 + 3 + f app(z', z'') -{ 1 }-> app(app(8, app(7, x)), app(7, y)) :|: x >= 0, y >= 0, z' = 7, z'' = 1 + (1 + 0 + x) + y app(z', z'') -{ 1 }-> app(app(8, app(0, y)), app(0, z)) :|: z >= 0, z' = 1 + 0 + (1 + (1 + 8 + y) + z), y >= 0, x >= 0, z'' = x app(z', z'') -{ 1 }-> app(app(8, app(0, y)), app(0, z)) :|: z'' = 1 + (1 + 8 + y) + z, z >= 0, x >= 0, y >= 0, z' = 1 + 0 + x app(z', z'') -{ 1 }-> app(app(8, app(0, y)), app(1 + 0 + x, z)) :|: z >= 0, z' = 1 + 0 + (1 + (1 + 8 + y) + z), y >= 0, x >= 0, z'' = x app(z', z'') -{ 1 }-> app(app(8, app(0, y)), app(1 + 0 + x, z)) :|: z'' = 1 + (1 + 8 + y) + z, z >= 0, x >= 0, y >= 0, z' = 1 + 0 + x app(z', z'') -{ 1 }-> app(app(8, app(1 + 0 + x, y)), app(0, z)) :|: z >= 0, z' = 1 + 0 + (1 + (1 + 8 + y) + z), y >= 0, x >= 0, z'' = x app(z', z'') -{ 1 }-> app(app(8, app(1 + 0 + x, y)), app(0, z)) :|: z'' = 1 + (1 + 8 + y) + z, z >= 0, x >= 0, y >= 0, z' = 1 + 0 + x app(z', z'') -{ 1 }-> app(app(8, app(1 + 0 + x, y)), app(1 + 0 + x, z)) :|: z >= 0, z' = 1 + 0 + (1 + (1 + 8 + y) + z), y >= 0, x >= 0, z'' = x app(z', z'') -{ 1 }-> app(app(8, app(1 + 0 + x, y)), app(1 + 0 + x, z)) :|: z'' = 1 + (1 + 8 + y) + z, z >= 0, x >= 0, y >= 0, z' = 1 + 0 + x app(z', z'') -{ 1 }-> app(app(1, app(f, x)), app(app(5, f), xs)) :|: f >= 0, xs >= 0, x >= 0, z' = 1 + 5 + f, z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 1 }-> app(app(0, app(7, x)), app(7, y)) :|: x >= 0, y >= 0, z' = 7, z'' = 1 + (1 + 8 + x) + y app(z', z'') -{ 1 }-> app(0, xs) :|: f >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 4 + 2) + f) + x app(z', z'') -{ 1 }-> app(0, app(0, xs)) :|: f >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 4 + 9) + f) + x app(z', z'') -{ 1 }-> app(0, app(1 + 3 + f, xs)) :|: f >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 4 + 9) + f) + x app(z', z'') -{ 1 }-> app(1 + 3 + f, xs) :|: f >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 4 + 2) + f) + x app(z', z'') -{ 1 }-> app(1 + 1 + x, app(0, xs)) :|: f >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 4 + 9) + f) + x app(z', z'') -{ 1 }-> app(1 + 1 + x, app(1 + 3 + f, xs)) :|: f >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 4 + 9) + f) + x app(z', z'') -{ 1 }-> 6 :|: f >= 0, z'' = 6, z' = 1 + 5 + f app(z', z'') -{ 1 }-> 6 :|: f >= 0, z'' = 6, z' = 1 + 3 + f app(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 app(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0 encArg(z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 9 :|: z' = 9 encArg(z') -{ 0 }-> 8 :|: z' = 8 encArg(z') -{ 0 }-> 7 :|: z' = 7 encArg(z') -{ 0 }-> 6 :|: z' = 6 encArg(z') -{ 0 }-> 5 :|: z' = 5 encArg(z') -{ 0 }-> 4 :|: z' = 4 encArg(z') -{ 0 }-> 3 :|: z' = 3 encArg(z') -{ 0 }-> 2 :|: z' = 2 encArg(z') -{ 0 }-> 1 :|: z' = 1 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_and -{ 0 }-> 0 :|: encode_app(z', z'') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_app(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 3 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 4 :|: encode_filter2 -{ 0 }-> 0 :|: encode_map -{ 0 }-> 5 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 6 :|: encode_nil -{ 0 }-> 0 :|: encode_not -{ 0 }-> 7 :|: encode_not -{ 0 }-> 0 :|: encode_or -{ 0 }-> 8 :|: encode_or -{ 0 }-> 0 :|: encode_true -{ 0 }-> 9 :|: encode_true -{ 0 }-> 0 :|: ---------------------------------------- (29) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 2 }-> app(app(app(app(4, x9), 7), 1 + 7 + x9), xs) :|: xs >= 0, z' = 1 + 3 + 7, z'' = 1 + (1 + 1 + (1 + 7 + x9)) + xs, x9 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(app(app(4, app(z' - 8, x12)), z' - 8), x12), xs2)), 1 + 3 + (z' - 8)), 1 + (1 + 1 + x12) + xs2), xs) :|: xs >= 0, xs2 >= 0, z' - 8 >= 0, x12 >= 0, z'' = 1 + (1 + 1 + (1 + (1 + 1 + x12) + xs2)) + xs app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(8, app(app(0, x), y3)), app(app(0, x), z1))), 1 + 0 + (1 + (1 + 8 + y3) + z1)), x), xs) :|: xs >= 0, z1 >= 0, x >= 0, y3 >= 0, z' = 1 + 3 + (1 + 0 + (1 + (1 + 8 + y3) + z1)), z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(8, app(app(0, z' - 5), y6)), app(app(0, z' - 5), z2))), 1 + 0 + (z' - 5)), 1 + (1 + 8 + y6) + z2), xs) :|: z'' = 1 + (1 + 1 + (1 + (1 + 8 + y6) + z2)) + xs, xs >= 0, y6 >= 0, z' - 5 >= 0, z2 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(8, app(7, x14)), app(7, y5))), 7), 1 + (1 + 0 + x14) + y5), xs) :|: y5 >= 0, xs >= 0, z' = 1 + 3 + 7, z'' = 1 + (1 + 1 + (1 + (1 + 0 + x14) + y5)) + xs, x14 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(3, f7), x)), 1 + (1 + (1 + 4 + 2) + f7) + x11), x), xs) :|: xs >= 0, z' = 1 + 3 + (1 + (1 + (1 + 4 + 2) + f7) + x11), x >= 0, f7 >= 0, x11 >= 0, z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(1, x10), app(app(3, f6), x))), 1 + (1 + (1 + 4 + 9) + f6) + x10), x), xs) :|: z' = 1 + 3 + (1 + (1 + (1 + 4 + 9) + f6) + x10), xs >= 0, x >= 0, f6 >= 0, x10 >= 0, z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(1, app(z' - 10, x8)), app(app(5, z' - 10), xs1))), 1 + 5 + (z' - 10)), 1 + (1 + 1 + x8) + xs1), xs) :|: xs >= 0, z'' = 1 + (1 + 1 + (1 + (1 + 1 + x8) + xs1)) + xs, x8 >= 0, xs1 >= 0, z' - 10 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, app(app(0, app(7, x13)), app(7, y4))), 7), 1 + (1 + 8 + x13) + y4), xs) :|: x13 >= 0, xs >= 0, z' = 1 + 3 + 7, z'' = 1 + (1 + 1 + (1 + (1 + 8 + x13) + y4)) + xs, y4 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, 6), 1 + 5 + (z' - 10)), 6), z'' - 9) :|: z' - 10 >= 0, z'' - 9 >= 0 app(z', z'') -{ 2 }-> app(app(app(app(4, 6), 1 + 3 + (z' - 8)), 6), z'' - 9) :|: z'' - 9 >= 0, z' - 8 >= 0 app(z', z'') -{ 1 }-> app(app(app(app(4, 0), z' - 4), x), xs) :|: z' - 4 >= 0, xs >= 0, x >= 0, z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 1 }-> app(app(app(app(4, 1 + (z' - 4) + x), z' - 4), x), xs) :|: z' - 4 >= 0, xs >= 0, x >= 0, z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 1 }-> app(app(8, app(7, x)), app(7, y)) :|: x >= 0, y >= 0, z' = 7, z'' = 1 + (1 + 0 + x) + y app(z', z'') -{ 1 }-> app(app(8, app(0, y)), app(0, z)) :|: z >= 0, z' = 1 + 0 + (1 + (1 + 8 + y) + z), y >= 0, z'' >= 0 app(z', z'') -{ 1 }-> app(app(8, app(0, y)), app(0, z)) :|: z'' = 1 + (1 + 8 + y) + z, z >= 0, z' - 1 >= 0, y >= 0 app(z', z'') -{ 1 }-> app(app(8, app(0, y)), app(1 + 0 + z'', z)) :|: z >= 0, z' = 1 + 0 + (1 + (1 + 8 + y) + z), y >= 0, z'' >= 0 app(z', z'') -{ 1 }-> app(app(8, app(0, y)), app(1 + 0 + (z' - 1), z)) :|: z'' = 1 + (1 + 8 + y) + z, z >= 0, z' - 1 >= 0, y >= 0 app(z', z'') -{ 1 }-> app(app(8, app(1 + 0 + z'', y)), app(0, z)) :|: z >= 0, z' = 1 + 0 + (1 + (1 + 8 + y) + z), y >= 0, z'' >= 0 app(z', z'') -{ 1 }-> app(app(8, app(1 + 0 + z'', y)), app(1 + 0 + z'', z)) :|: z >= 0, z' = 1 + 0 + (1 + (1 + 8 + y) + z), y >= 0, z'' >= 0 app(z', z'') -{ 1 }-> app(app(8, app(1 + 0 + (z' - 1), y)), app(0, z)) :|: z'' = 1 + (1 + 8 + y) + z, z >= 0, z' - 1 >= 0, y >= 0 app(z', z'') -{ 1 }-> app(app(8, app(1 + 0 + (z' - 1), y)), app(1 + 0 + (z' - 1), z)) :|: z'' = 1 + (1 + 8 + y) + z, z >= 0, z' - 1 >= 0, y >= 0 app(z', z'') -{ 1 }-> app(app(1, app(z' - 6, x)), app(app(5, z' - 6), xs)) :|: z' - 6 >= 0, xs >= 0, x >= 0, z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 1 }-> app(app(0, app(7, x)), app(7, y)) :|: x >= 0, y >= 0, z' = 7, z'' = 1 + (1 + 8 + x) + y app(z', z'') -{ 1 }-> app(0, z'') :|: f >= 0, z'' >= 0, x >= 0, z' = 1 + (1 + (1 + 4 + 2) + f) + x app(z', z'') -{ 1 }-> app(0, app(0, z'')) :|: f >= 0, z'' >= 0, x >= 0, z' = 1 + (1 + (1 + 4 + 9) + f) + x app(z', z'') -{ 1 }-> app(0, app(1 + 3 + f, z'')) :|: f >= 0, z'' >= 0, x >= 0, z' = 1 + (1 + (1 + 4 + 9) + f) + x app(z', z'') -{ 1 }-> app(1 + 3 + f, z'') :|: f >= 0, z'' >= 0, x >= 0, z' = 1 + (1 + (1 + 4 + 2) + f) + x app(z', z'') -{ 1 }-> app(1 + 1 + x, app(0, z'')) :|: f >= 0, z'' >= 0, x >= 0, z' = 1 + (1 + (1 + 4 + 9) + f) + x app(z', z'') -{ 1 }-> app(1 + 1 + x, app(1 + 3 + f, z'')) :|: f >= 0, z'' >= 0, x >= 0, z' = 1 + (1 + (1 + 4 + 9) + f) + x app(z', z'') -{ 1 }-> 6 :|: z' - 6 >= 0, z'' = 6 app(z', z'') -{ 1 }-> 6 :|: z' - 4 >= 0, z'' = 6 app(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 app(z', z'') -{ 1 }-> z'' - 8 :|: z'' - 8 >= 0, z' = 7 app(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 encArg(z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 9 :|: z' = 9 encArg(z') -{ 0 }-> 8 :|: z' = 8 encArg(z') -{ 0 }-> 7 :|: z' = 7 encArg(z') -{ 0 }-> 6 :|: z' = 6 encArg(z') -{ 0 }-> 5 :|: z' = 5 encArg(z') -{ 0 }-> 4 :|: z' = 4 encArg(z') -{ 0 }-> 3 :|: z' = 3 encArg(z') -{ 0 }-> 2 :|: z' = 2 encArg(z') -{ 0 }-> 1 :|: z' = 1 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encode_and -{ 0 }-> 0 :|: encode_app(z', z'') -{ 0 }-> app(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_app(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 3 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 4 :|: encode_filter2 -{ 0 }-> 0 :|: encode_map -{ 0 }-> 5 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 6 :|: encode_nil -{ 0 }-> 0 :|: encode_not -{ 0 }-> 7 :|: encode_not -{ 0 }-> 0 :|: encode_or -{ 0 }-> 8 :|: encode_or -{ 0 }-> 0 :|: encode_true -{ 0 }-> 9 :|: encode_true -{ 0 }-> 0 :|: ---------------------------------------- (31) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_not -> null_encode_not [0] encode_or -> null_encode_or [0] encode_and -> null_encode_and [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_not, null_encode_or, null_encode_and, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(and, c_app(c_app(or, y), z)), x) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) [1] app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(not, c_app(not, x)) -> x [1] app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) [1] app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs) [1] app(c_app(map, f), nil) -> nil [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) [1] app(not, c_app(c_app(or, x), y)) -> app(app(and, app(not, x)), app(not, y)) [1] app(not, c_app(c_app(and, x), y)) -> app(app(or, app(not, x)), app(not, y)) [1] app(c_app(and, x), c_app(c_app(or, y), z)) -> app(app(or, app(app(and, x), y)), app(app(and, x), z)) [1] encArg(not) -> not [0] encArg(or) -> or [0] encArg(and) -> and [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_not -> not [0] encode_or -> or [0] encode_and -> and [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_not -> null_encode_not [0] encode_or -> null_encode_or [0] encode_and -> null_encode_and [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app c_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encArg :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_not :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_or :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_and :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_map :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_nil :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_cons :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter2 :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_true :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_false :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_app :: and:or:c_app:map:cons:not:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_not:null_encode_or:null_encode_and:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: and => 0 or => 8 map => 5 cons => 1 not => 7 filter2 => 4 true => 9 filter => 3 false => 2 nil => 6 null_encArg => 0 null_encode_app => 0 null_encode_not => 0 null_encode_or => 0 null_encode_and => 0 null_encode_map => 0 null_encode_nil => 0 null_encode_cons => 0 null_encode_filter => 0 null_encode_filter2 => 0 null_encode_true => 0 null_encode_false => 0 null_app => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> x :|: z'' = 1 + 7 + x, x >= 0, z' = 7 app(z', z'') -{ 1 }-> app(app(app(app(4, app(f, x)), f), x), xs) :|: f >= 0, xs >= 0, x >= 0, z'' = 1 + (1 + 1 + x) + xs, z' = 1 + 3 + f app(z', z'') -{ 1 }-> app(app(8, app(app(0, x), y)), app(app(0, x), z)) :|: z >= 0, z' = 1 + 0 + (1 + (1 + 8 + y) + z), y >= 0, x >= 0, z'' = x app(z', z'') -{ 1 }-> app(app(8, app(app(0, x), y)), app(app(0, x), z)) :|: z'' = 1 + (1 + 8 + y) + z, z >= 0, x >= 0, y >= 0, z' = 1 + 0 + x app(z', z'') -{ 1 }-> app(app(8, app(7, x)), app(7, y)) :|: x >= 0, y >= 0, z' = 7, z'' = 1 + (1 + 0 + x) + y app(z', z'') -{ 1 }-> app(app(3, f), xs) :|: f >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 4 + 2) + f) + x app(z', z'') -{ 1 }-> app(app(1, x), app(app(3, f), xs)) :|: f >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 4 + 9) + f) + x app(z', z'') -{ 1 }-> app(app(1, app(f, x)), app(app(5, f), xs)) :|: f >= 0, xs >= 0, x >= 0, z' = 1 + 5 + f, z'' = 1 + (1 + 1 + x) + xs app(z', z'') -{ 1 }-> app(app(0, app(7, x)), app(7, y)) :|: x >= 0, y >= 0, z' = 7, z'' = 1 + (1 + 8 + x) + y app(z', z'') -{ 1 }-> 6 :|: f >= 0, z'' = 6, z' = 1 + 5 + f app(z', z'') -{ 1 }-> 6 :|: f >= 0, z'' = 6, z' = 1 + 3 + f app(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 app(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0 encArg(z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 9 :|: z' = 9 encArg(z') -{ 0 }-> 8 :|: z' = 8 encArg(z') -{ 0 }-> 7 :|: z' = 7 encArg(z') -{ 0 }-> 6 :|: z' = 6 encArg(z') -{ 0 }-> 5 :|: z' = 5 encArg(z') -{ 0 }-> 4 :|: z' = 4 encArg(z') -{ 0 }-> 3 :|: z' = 3 encArg(z') -{ 0 }-> 2 :|: z' = 2 encArg(z') -{ 0 }-> 1 :|: z' = 1 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_and -{ 0 }-> 0 :|: encode_app(z', z'') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_app(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 3 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 4 :|: encode_filter2 -{ 0 }-> 0 :|: encode_map -{ 0 }-> 5 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 6 :|: encode_nil -{ 0 }-> 0 :|: encode_not -{ 0 }-> 7 :|: encode_not -{ 0 }-> 0 :|: encode_or -{ 0 }-> 8 :|: encode_or -{ 0 }-> 0 :|: encode_true -{ 0 }-> 9 :|: encode_true -{ 0 }-> 0 :|: Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(not) -> c ENCARG(or) -> c1 ENCARG(and) -> c2 ENCARG(map) -> c3 ENCARG(nil) -> c4 ENCARG(cons) -> c5 ENCARG(filter) -> c6 ENCARG(filter2) -> c7 ENCARG(true) -> c8 ENCARG(false) -> c9 ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_NOT -> c12 ENCODE_OR -> c13 ENCODE_AND -> c14 ENCODE_MAP -> c15 ENCODE_NIL -> c16 ENCODE_CONS -> c17 ENCODE_FILTER -> c18 ENCODE_FILTER2 -> c19 ENCODE_TRUE -> c20 ENCODE_FALSE -> c21 APP(not, app(not, z0)) -> c22 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(and, app(not, z0)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(app(or, app(not, z0)), app(not, z1)), APP(or, app(not, z0)), APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)), APP(or, app(app(and, z0), z1)), APP(app(and, z0), z1), APP(and, z0), APP(app(and, z0), z2), APP(and, z0)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)), APP(or, app(app(and, z2), z0)), APP(app(and, z2), z0), APP(and, z2), APP(app(and, z2), z1), APP(and, z2)) APP(app(map, z0), nil) -> c27 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0)) APP(app(filter, z0), nil) -> c29 APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(not, app(not, z0)) -> c22 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(and, app(not, z0)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(app(or, app(not, z0)), app(not, z1)), APP(or, app(not, z0)), APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)), APP(or, app(app(and, z0), z1)), APP(app(and, z0), z1), APP(and, z0), APP(app(and, z0), z2), APP(and, z0)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)), APP(or, app(app(and, z2), z0)), APP(app(and, z2), z0), APP(and, z2), APP(app(and, z2), z1), APP(and, z2)) APP(app(map, z0), nil) -> c27 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0)) APP(app(filter, z0), nil) -> c29 APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_not, encode_or, encode_and, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, ENCODE_NOT, ENCODE_OR, ENCODE_AND, ENCODE_MAP, ENCODE_NIL, ENCODE_CONS, ENCODE_FILTER, ENCODE_FILTER2, ENCODE_TRUE, ENCODE_FALSE, APP_2 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10_3, c11_3, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23_4, c24_4, c25_6, c26_6, c27, c28_5, c29, c30_5, c31_4, c32_2 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 20 trailing nodes: ENCARG(nil) -> c4 ENCODE_NOT -> c12 ENCODE_TRUE -> c20 ENCODE_AND -> c14 ENCARG(map) -> c3 ENCODE_OR -> c13 ENCARG(filter) -> c6 ENCARG(false) -> c9 ENCODE_NIL -> c16 ENCARG(or) -> c1 ENCODE_FILTER -> c18 ENCARG(and) -> c2 ENCARG(true) -> c8 ENCODE_CONS -> c17 ENCODE_MAP -> c15 ENCODE_FALSE -> c21 ENCARG(filter2) -> c7 ENCODE_FILTER2 -> c19 ENCARG(cons) -> c5 ENCARG(not) -> c ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(not, app(not, z0)) -> c22 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(and, app(not, z0)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(app(or, app(not, z0)), app(not, z1)), APP(or, app(not, z0)), APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)), APP(or, app(app(and, z0), z1)), APP(app(and, z0), z1), APP(and, z0), APP(app(and, z0), z2), APP(and, z0)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)), APP(or, app(app(and, z2), z0)), APP(app(and, z2), z0), APP(and, z2), APP(app(and, z2), z1), APP(and, z2)) APP(app(map, z0), nil) -> c27 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0)) APP(app(filter, z0), nil) -> c29 APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(not, app(not, z0)) -> c22 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(and, app(not, z0)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(app(or, app(not, z0)), app(not, z1)), APP(or, app(not, z0)), APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)), APP(or, app(app(and, z0), z1)), APP(app(and, z0), z1), APP(and, z0), APP(app(and, z0), z2), APP(and, z0)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)), APP(or, app(app(and, z2), z0)), APP(app(and, z2), z0), APP(and, z2), APP(app(and, z2), z1), APP(and, z2)) APP(app(map, z0), nil) -> c27 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0)) APP(app(filter, z0), nil) -> c29 APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_not, encode_or, encode_and, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c10_3, c11_3, c22, c23_4, c24_4, c25_6, c26_6, c27, c28_5, c29, c30_5, c31_4, c32_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 21 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_not, encode_or, encode_and, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c10_3, c11_3, c22, c27, c29, c23_3, c24_2, c25_2, c26_2, c28_2, c30_2, c31_1, c32_1 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCODE_APP(z0, z1) -> c(ENCARG(z0)) ENCODE_APP(z0, z1) -> c(ENCARG(z1)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_not, encode_or, encode_and, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c10_3, c22, c27, c29, c23_3, c24_2, c25_2, c26_2, c28_2, c30_2, c31_1, c32_1, c_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_APP(z0, z1) -> c(ENCARG(z0)) ENCODE_APP(z0, z1) -> c(ENCARG(z1)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_not, encode_or, encode_and, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c10_3, c22, c27, c29, c23_3, c24_2, c25_2, c26_2, c28_2, c30_2, c31_1, c32_1, c_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_not -> not encode_or -> or encode_and -> and encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c10_3, c22, c27, c29, c23_3, c24_2, c25_2, c26_2, c28_2, c30_2, c31_1, c32_1, c_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0), ENCARG(not)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0), ENCARG(or)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0), ENCARG(and)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(not), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(APP(or, encArg(x1)), ENCARG(or), ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(APP(and, encArg(x1)), ENCARG(and), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0), ENCARG(not)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0), ENCARG(or)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0), ENCARG(and)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(not), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(APP(or, encArg(x1)), ENCARG(or), ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(APP(and, encArg(x1)), ENCARG(and), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22, c27, c29, c23_3, c24_2, c25_2, c26_2, c28_2, c30_2, c31_1, c32_1, c_1, c10_3 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 29 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22, c27, c29, c23_3, c24_2, c25_2, c26_2, c28_2, c30_2, c31_1, c32_1, c_1, c10_3, c10_2, c10_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(not, app(app(or, z0), z1)) -> c23(APP(app(and, app(not, z0)), app(not, z1)), APP(not, z0), APP(not, z1)) by APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c30_2, c31_1, c32_1, c_1, c10_3, c10_2, c10_1, c23_3 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) by APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(or, app(not, z0)), app(not, z1))), not), app(app(and, z0), z1)), x2), APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2))), app(and, z0)), app(app(or, z1), z2)), x2), APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1))), app(and, app(app(or, z0), z1))), z2), x2), APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(map, z0)), nil), x2), APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2), APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(filter, z0)), nil), x2), APP(app(filter, z0), nil)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2), APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(or, app(not, z0)), app(not, z1))), not), app(app(and, z0), z1)), x2), APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2))), app(and, z0)), app(app(or, z1), z2)), x2), APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1))), app(and, app(app(or, z0), z1))), z2), x2), APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(map, z0)), nil), x2), APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2), APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(filter, z0)), nil), x2), APP(app(filter, z0), nil)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2), APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(or, app(not, z0)), app(not, z1))), not), app(app(and, z0), z1)), x2), APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2))), app(and, z0)), app(app(or, z1), z2)), x2), APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1))), app(and, app(app(or, z0), z1))), z2), x2), APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(map, z0)), nil), x2), APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2), APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(filter, z0)), nil), x2), APP(app(filter, z0), nil)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2), APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c_1, c10_3, c10_2, c10_1, c23_3, c30_2 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c_1, c10_3, c10_2, c10_1, c23_3, c30_2, c30_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) by ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(or, x1) -> c(APP(or, encArg(x1))) ENCODE_APP(and, x1) -> c(APP(and, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(or, x1) -> c(APP(or, encArg(x1))) ENCODE_APP(and, x1) -> c(APP(and, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_3, c10_2, c10_1, c23_3, c30_2, c30_1, c_1 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(or, x1) -> c(APP(or, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(and, x1) -> c(APP(and, encArg(x1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_3, c10_2, c10_1, c23_3, c30_2, c30_1, c_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(not), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(APP(or, app(encArg(x1), encArg(x2))), ENCARG(or), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(APP(and, app(encArg(x1), encArg(x2))), ENCARG(and), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(not), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(APP(or, app(encArg(x1), encArg(x2))), ENCARG(or), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(APP(and, app(encArg(x1), encArg(x2))), ENCARG(and), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_3, c10_2, c10_1, c23_3, c30_2, c30_1, c_1 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 19 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_3, c10_2, c10_1, c23_3, c30_2, c30_1, c_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1)), ENCARG(not)) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1)), ENCARG(or)) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1)), ENCARG(and)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, x1), cons_app(z0, z1))) -> c10(APP(app(encArg(x0), encArg(x1)), app(encArg(z0), encArg(z1))), ENCARG(cons_app(x0, x1)), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(APP(app(or, encArg(x1)), encArg(x2)), ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1)), ENCARG(not)) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1)), ENCARG(or)) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1)), ENCARG(and)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(APP(app(or, encArg(x1)), encArg(x2)), ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, not)) -> c10(APP(encArg(x0), not), ENCARG(x0)) by ENCARG(cons_app(not, not)) -> c10(APP(not, not), ENCARG(not)) ENCARG(cons_app(or, not)) -> c10(APP(or, not), ENCARG(or)) ENCARG(cons_app(and, not)) -> c10(APP(and, not), ENCARG(and)) ENCARG(cons_app(map, not)) -> c10(APP(map, not), ENCARG(map)) ENCARG(cons_app(nil, not)) -> c10(APP(nil, not), ENCARG(nil)) ENCARG(cons_app(cons, not)) -> c10(APP(cons, not), ENCARG(cons)) ENCARG(cons_app(filter, not)) -> c10(APP(filter, not), ENCARG(filter)) ENCARG(cons_app(filter2, not)) -> c10(APP(filter2, not), ENCARG(filter2)) ENCARG(cons_app(true, not)) -> c10(APP(true, not), ENCARG(true)) ENCARG(cons_app(false, not)) -> c10(APP(false, not), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), not)) -> c10(APP(app(encArg(z0), encArg(z1)), not), ENCARG(cons_app(z0, z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, not)) -> c10(APP(not, not), ENCARG(not)) ENCARG(cons_app(or, not)) -> c10(APP(or, not), ENCARG(or)) ENCARG(cons_app(and, not)) -> c10(APP(and, not), ENCARG(and)) ENCARG(cons_app(map, not)) -> c10(APP(map, not), ENCARG(map)) ENCARG(cons_app(nil, not)) -> c10(APP(nil, not), ENCARG(nil)) ENCARG(cons_app(cons, not)) -> c10(APP(cons, not), ENCARG(cons)) ENCARG(cons_app(filter, not)) -> c10(APP(filter, not), ENCARG(filter)) ENCARG(cons_app(filter2, not)) -> c10(APP(filter2, not), ENCARG(filter2)) ENCARG(cons_app(true, not)) -> c10(APP(true, not), ENCARG(true)) ENCARG(cons_app(false, not)) -> c10(APP(false, not), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(true, not)) -> c10(APP(true, not), ENCARG(true)) ENCARG(cons_app(nil, not)) -> c10(APP(nil, not), ENCARG(nil)) ENCARG(cons_app(cons, not)) -> c10(APP(cons, not), ENCARG(cons)) ENCARG(cons_app(and, not)) -> c10(APP(and, not), ENCARG(and)) ENCARG(cons_app(false, not)) -> c10(APP(false, not), ENCARG(false)) ENCARG(cons_app(filter2, not)) -> c10(APP(filter2, not), ENCARG(filter2)) ENCARG(cons_app(map, not)) -> c10(APP(map, not), ENCARG(map)) ENCARG(cons_app(filter, not)) -> c10(APP(filter, not), ENCARG(filter)) ENCARG(cons_app(or, not)) -> c10(APP(or, not), ENCARG(or)) ENCARG(cons_app(not, not)) -> c10(APP(not, not), ENCARG(not)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, or)) -> c10(APP(encArg(x0), or), ENCARG(x0)) by ENCARG(cons_app(not, or)) -> c10(APP(not, or), ENCARG(not)) ENCARG(cons_app(or, or)) -> c10(APP(or, or), ENCARG(or)) ENCARG(cons_app(and, or)) -> c10(APP(and, or), ENCARG(and)) ENCARG(cons_app(map, or)) -> c10(APP(map, or), ENCARG(map)) ENCARG(cons_app(nil, or)) -> c10(APP(nil, or), ENCARG(nil)) ENCARG(cons_app(cons, or)) -> c10(APP(cons, or), ENCARG(cons)) ENCARG(cons_app(filter, or)) -> c10(APP(filter, or), ENCARG(filter)) ENCARG(cons_app(filter2, or)) -> c10(APP(filter2, or), ENCARG(filter2)) ENCARG(cons_app(true, or)) -> c10(APP(true, or), ENCARG(true)) ENCARG(cons_app(false, or)) -> c10(APP(false, or), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), or)) -> c10(APP(app(encArg(z0), encArg(z1)), or), ENCARG(cons_app(z0, z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, or)) -> c10(APP(not, or), ENCARG(not)) ENCARG(cons_app(or, or)) -> c10(APP(or, or), ENCARG(or)) ENCARG(cons_app(and, or)) -> c10(APP(and, or), ENCARG(and)) ENCARG(cons_app(map, or)) -> c10(APP(map, or), ENCARG(map)) ENCARG(cons_app(nil, or)) -> c10(APP(nil, or), ENCARG(nil)) ENCARG(cons_app(cons, or)) -> c10(APP(cons, or), ENCARG(cons)) ENCARG(cons_app(filter, or)) -> c10(APP(filter, or), ENCARG(filter)) ENCARG(cons_app(filter2, or)) -> c10(APP(filter2, or), ENCARG(filter2)) ENCARG(cons_app(true, or)) -> c10(APP(true, or), ENCARG(true)) ENCARG(cons_app(false, or)) -> c10(APP(false, or), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(false, or)) -> c10(APP(false, or), ENCARG(false)) ENCARG(cons_app(and, or)) -> c10(APP(and, or), ENCARG(and)) ENCARG(cons_app(or, or)) -> c10(APP(or, or), ENCARG(or)) ENCARG(cons_app(nil, or)) -> c10(APP(nil, or), ENCARG(nil)) ENCARG(cons_app(filter2, or)) -> c10(APP(filter2, or), ENCARG(filter2)) ENCARG(cons_app(map, or)) -> c10(APP(map, or), ENCARG(map)) ENCARG(cons_app(cons, or)) -> c10(APP(cons, or), ENCARG(cons)) ENCARG(cons_app(not, or)) -> c10(APP(not, or), ENCARG(not)) ENCARG(cons_app(true, or)) -> c10(APP(true, or), ENCARG(true)) ENCARG(cons_app(filter, or)) -> c10(APP(filter, or), ENCARG(filter)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, and)) -> c10(APP(encArg(x0), and), ENCARG(x0)) by ENCARG(cons_app(not, and)) -> c10(APP(not, and), ENCARG(not)) ENCARG(cons_app(or, and)) -> c10(APP(or, and), ENCARG(or)) ENCARG(cons_app(and, and)) -> c10(APP(and, and), ENCARG(and)) ENCARG(cons_app(map, and)) -> c10(APP(map, and), ENCARG(map)) ENCARG(cons_app(nil, and)) -> c10(APP(nil, and), ENCARG(nil)) ENCARG(cons_app(cons, and)) -> c10(APP(cons, and), ENCARG(cons)) ENCARG(cons_app(filter, and)) -> c10(APP(filter, and), ENCARG(filter)) ENCARG(cons_app(filter2, and)) -> c10(APP(filter2, and), ENCARG(filter2)) ENCARG(cons_app(true, and)) -> c10(APP(true, and), ENCARG(true)) ENCARG(cons_app(false, and)) -> c10(APP(false, and), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), and)) -> c10(APP(app(encArg(z0), encArg(z1)), and), ENCARG(cons_app(z0, z1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, and)) -> c10(APP(not, and), ENCARG(not)) ENCARG(cons_app(or, and)) -> c10(APP(or, and), ENCARG(or)) ENCARG(cons_app(and, and)) -> c10(APP(and, and), ENCARG(and)) ENCARG(cons_app(map, and)) -> c10(APP(map, and), ENCARG(map)) ENCARG(cons_app(nil, and)) -> c10(APP(nil, and), ENCARG(nil)) ENCARG(cons_app(cons, and)) -> c10(APP(cons, and), ENCARG(cons)) ENCARG(cons_app(filter, and)) -> c10(APP(filter, and), ENCARG(filter)) ENCARG(cons_app(filter2, and)) -> c10(APP(filter2, and), ENCARG(filter2)) ENCARG(cons_app(true, and)) -> c10(APP(true, and), ENCARG(true)) ENCARG(cons_app(false, and)) -> c10(APP(false, and), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(filter2, and)) -> c10(APP(filter2, and), ENCARG(filter2)) ENCARG(cons_app(map, and)) -> c10(APP(map, and), ENCARG(map)) ENCARG(cons_app(cons, and)) -> c10(APP(cons, and), ENCARG(cons)) ENCARG(cons_app(and, and)) -> c10(APP(and, and), ENCARG(and)) ENCARG(cons_app(nil, and)) -> c10(APP(nil, and), ENCARG(nil)) ENCARG(cons_app(or, and)) -> c10(APP(or, and), ENCARG(or)) ENCARG(cons_app(false, and)) -> c10(APP(false, and), ENCARG(false)) ENCARG(cons_app(true, and)) -> c10(APP(true, and), ENCARG(true)) ENCARG(cons_app(not, and)) -> c10(APP(not, and), ENCARG(not)) ENCARG(cons_app(filter, and)) -> c10(APP(filter, and), ENCARG(filter)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) by ENCARG(cons_app(not, map)) -> c10(APP(not, map), ENCARG(not)) ENCARG(cons_app(or, map)) -> c10(APP(or, map), ENCARG(or)) ENCARG(cons_app(and, map)) -> c10(APP(and, map), ENCARG(and)) ENCARG(cons_app(map, map)) -> c10(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c10(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c10(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c10(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c10(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c10(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c10(APP(false, map), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), map)) -> c10(APP(app(encArg(z0), encArg(z1)), map), ENCARG(cons_app(z0, z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, map)) -> c10(APP(not, map), ENCARG(not)) ENCARG(cons_app(or, map)) -> c10(APP(or, map), ENCARG(or)) ENCARG(cons_app(and, map)) -> c10(APP(and, map), ENCARG(and)) ENCARG(cons_app(map, map)) -> c10(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c10(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c10(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c10(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c10(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c10(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c10(APP(false, map), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(and, map)) -> c10(APP(and, map), ENCARG(and)) ENCARG(cons_app(nil, map)) -> c10(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(true, map)) -> c10(APP(true, map), ENCARG(true)) ENCARG(cons_app(cons, map)) -> c10(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter2, map)) -> c10(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(map, map)) -> c10(APP(map, map), ENCARG(map)) ENCARG(cons_app(not, map)) -> c10(APP(not, map), ENCARG(not)) ENCARG(cons_app(false, map)) -> c10(APP(false, map), ENCARG(false)) ENCARG(cons_app(filter, map)) -> c10(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(or, map)) -> c10(APP(or, map), ENCARG(or)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) by ENCARG(cons_app(not, nil)) -> c10(APP(not, nil), ENCARG(not)) ENCARG(cons_app(or, nil)) -> c10(APP(or, nil), ENCARG(or)) ENCARG(cons_app(and, nil)) -> c10(APP(and, nil), ENCARG(and)) ENCARG(cons_app(map, nil)) -> c10(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c10(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c10(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c10(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c10(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c10(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c10(APP(false, nil), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), nil)) -> c10(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, nil)) -> c10(APP(not, nil), ENCARG(not)) ENCARG(cons_app(or, nil)) -> c10(APP(or, nil), ENCARG(or)) ENCARG(cons_app(and, nil)) -> c10(APP(and, nil), ENCARG(and)) ENCARG(cons_app(map, nil)) -> c10(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c10(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c10(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c10(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c10(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c10(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c10(APP(false, nil), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(filter2, nil)) -> c10(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(map, nil)) -> c10(APP(map, nil), ENCARG(map)) ENCARG(cons_app(and, nil)) -> c10(APP(and, nil), ENCARG(and)) ENCARG(cons_app(nil, nil)) -> c10(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(or, nil)) -> c10(APP(or, nil), ENCARG(or)) ENCARG(cons_app(cons, nil)) -> c10(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(false, nil)) -> c10(APP(false, nil), ENCARG(false)) ENCARG(cons_app(filter, nil)) -> c10(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(true, nil)) -> c10(APP(true, nil), ENCARG(true)) ENCARG(cons_app(not, nil)) -> c10(APP(not, nil), ENCARG(not)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) by ENCARG(cons_app(not, cons)) -> c10(APP(not, cons), ENCARG(not)) ENCARG(cons_app(or, cons)) -> c10(APP(or, cons), ENCARG(or)) ENCARG(cons_app(and, cons)) -> c10(APP(and, cons), ENCARG(and)) ENCARG(cons_app(map, cons)) -> c10(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c10(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c10(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c10(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c10(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c10(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c10(APP(false, cons), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), cons)) -> c10(APP(app(encArg(z0), encArg(z1)), cons), ENCARG(cons_app(z0, z1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, cons)) -> c10(APP(not, cons), ENCARG(not)) ENCARG(cons_app(or, cons)) -> c10(APP(or, cons), ENCARG(or)) ENCARG(cons_app(and, cons)) -> c10(APP(and, cons), ENCARG(and)) ENCARG(cons_app(map, cons)) -> c10(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c10(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c10(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c10(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c10(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c10(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c10(APP(false, cons), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(false, cons)) -> c10(APP(false, cons), ENCARG(false)) ENCARG(cons_app(cons, cons)) -> c10(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c10(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(and, cons)) -> c10(APP(and, cons), ENCARG(and)) ENCARG(cons_app(filter2, cons)) -> c10(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(map, cons)) -> c10(APP(map, cons), ENCARG(map)) ENCARG(cons_app(true, cons)) -> c10(APP(true, cons), ENCARG(true)) ENCARG(cons_app(or, cons)) -> c10(APP(or, cons), ENCARG(or)) ENCARG(cons_app(nil, cons)) -> c10(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(not, cons)) -> c10(APP(not, cons), ENCARG(not)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) by ENCARG(cons_app(not, filter)) -> c10(APP(not, filter), ENCARG(not)) ENCARG(cons_app(or, filter)) -> c10(APP(or, filter), ENCARG(or)) ENCARG(cons_app(and, filter)) -> c10(APP(and, filter), ENCARG(and)) ENCARG(cons_app(map, filter)) -> c10(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c10(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c10(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c10(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c10(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c10(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c10(APP(false, filter), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter)) -> c10(APP(app(encArg(z0), encArg(z1)), filter), ENCARG(cons_app(z0, z1))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, filter)) -> c10(APP(not, filter), ENCARG(not)) ENCARG(cons_app(or, filter)) -> c10(APP(or, filter), ENCARG(or)) ENCARG(cons_app(and, filter)) -> c10(APP(and, filter), ENCARG(and)) ENCARG(cons_app(map, filter)) -> c10(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c10(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c10(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c10(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c10(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c10(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c10(APP(false, filter), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(not, filter)) -> c10(APP(not, filter), ENCARG(not)) ENCARG(cons_app(filter, filter)) -> c10(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(cons, filter)) -> c10(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(nil, filter)) -> c10(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(and, filter)) -> c10(APP(and, filter), ENCARG(and)) ENCARG(cons_app(or, filter)) -> c10(APP(or, filter), ENCARG(or)) ENCARG(cons_app(filter2, filter)) -> c10(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(map, filter)) -> c10(APP(map, filter), ENCARG(map)) ENCARG(cons_app(true, filter)) -> c10(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c10(APP(false, filter), ENCARG(false)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) by ENCARG(cons_app(not, filter2)) -> c10(APP(not, filter2), ENCARG(not)) ENCARG(cons_app(or, filter2)) -> c10(APP(or, filter2), ENCARG(or)) ENCARG(cons_app(and, filter2)) -> c10(APP(and, filter2), ENCARG(and)) ENCARG(cons_app(map, filter2)) -> c10(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c10(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c10(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c10(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c10(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c10(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c10(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter2)) -> c10(APP(app(encArg(z0), encArg(z1)), filter2), ENCARG(cons_app(z0, z1))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, filter2)) -> c10(APP(not, filter2), ENCARG(not)) ENCARG(cons_app(or, filter2)) -> c10(APP(or, filter2), ENCARG(or)) ENCARG(cons_app(and, filter2)) -> c10(APP(and, filter2), ENCARG(and)) ENCARG(cons_app(map, filter2)) -> c10(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c10(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c10(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c10(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c10(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c10(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c10(APP(false, filter2), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(filter, filter2)) -> c10(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(cons, filter2)) -> c10(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(nil, filter2)) -> c10(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(false, filter2)) -> c10(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(and, filter2)) -> c10(APP(and, filter2), ENCARG(and)) ENCARG(cons_app(filter2, filter2)) -> c10(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(map, filter2)) -> c10(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(or, filter2)) -> c10(APP(or, filter2), ENCARG(or)) ENCARG(cons_app(not, filter2)) -> c10(APP(not, filter2), ENCARG(not)) ENCARG(cons_app(true, filter2)) -> c10(APP(true, filter2), ENCARG(true)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) by ENCARG(cons_app(not, true)) -> c10(APP(not, true), ENCARG(not)) ENCARG(cons_app(or, true)) -> c10(APP(or, true), ENCARG(or)) ENCARG(cons_app(and, true)) -> c10(APP(and, true), ENCARG(and)) ENCARG(cons_app(map, true)) -> c10(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c10(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c10(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c10(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c10(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c10(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c10(APP(false, true), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), true)) -> c10(APP(app(encArg(z0), encArg(z1)), true), ENCARG(cons_app(z0, z1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, true)) -> c10(APP(not, true), ENCARG(not)) ENCARG(cons_app(or, true)) -> c10(APP(or, true), ENCARG(or)) ENCARG(cons_app(and, true)) -> c10(APP(and, true), ENCARG(and)) ENCARG(cons_app(map, true)) -> c10(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c10(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c10(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c10(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c10(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c10(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c10(APP(false, true), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(nil, true)) -> c10(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(and, true)) -> c10(APP(and, true), ENCARG(and)) ENCARG(cons_app(or, true)) -> c10(APP(or, true), ENCARG(or)) ENCARG(cons_app(filter, true)) -> c10(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(not, true)) -> c10(APP(not, true), ENCARG(not)) ENCARG(cons_app(filter2, true)) -> c10(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(map, true)) -> c10(APP(map, true), ENCARG(map)) ENCARG(cons_app(false, true)) -> c10(APP(false, true), ENCARG(false)) ENCARG(cons_app(true, true)) -> c10(APP(true, true), ENCARG(true)) ENCARG(cons_app(cons, true)) -> c10(APP(cons, true), ENCARG(cons)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) by ENCARG(cons_app(not, false)) -> c10(APP(not, false), ENCARG(not)) ENCARG(cons_app(or, false)) -> c10(APP(or, false), ENCARG(or)) ENCARG(cons_app(and, false)) -> c10(APP(and, false), ENCARG(and)) ENCARG(cons_app(map, false)) -> c10(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c10(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c10(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c10(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c10(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c10(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c10(APP(false, false), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), false)) -> c10(APP(app(encArg(z0), encArg(z1)), false), ENCARG(cons_app(z0, z1))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, false)) -> c10(APP(not, false), ENCARG(not)) ENCARG(cons_app(or, false)) -> c10(APP(or, false), ENCARG(or)) ENCARG(cons_app(and, false)) -> c10(APP(and, false), ENCARG(and)) ENCARG(cons_app(map, false)) -> c10(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c10(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c10(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c10(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c10(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c10(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c10(APP(false, false), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(filter, false)) -> c10(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(nil, false)) -> c10(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(and, false)) -> c10(APP(and, false), ENCARG(and)) ENCARG(cons_app(false, false)) -> c10(APP(false, false), ENCARG(false)) ENCARG(cons_app(true, false)) -> c10(APP(true, false), ENCARG(true)) ENCARG(cons_app(cons, false)) -> c10(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(or, false)) -> c10(APP(or, false), ENCARG(or)) ENCARG(cons_app(filter2, false)) -> c10(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(map, false)) -> c10(APP(map, false), ENCARG(map)) ENCARG(cons_app(not, false)) -> c10(APP(not, false), ENCARG(not)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_2, c10_1, c23_3, c30_2, c30_1, c_1, c10_3 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(not, x1)) -> c10(APP(not, encArg(x1)), ENCARG(x1)) by ENCARG(cons_app(not, not)) -> c10(APP(not, not), ENCARG(not)) ENCARG(cons_app(not, or)) -> c10(APP(not, or), ENCARG(or)) ENCARG(cons_app(not, and)) -> c10(APP(not, and), ENCARG(and)) ENCARG(cons_app(not, map)) -> c10(APP(not, map), ENCARG(map)) ENCARG(cons_app(not, nil)) -> c10(APP(not, nil), ENCARG(nil)) ENCARG(cons_app(not, cons)) -> c10(APP(not, cons), ENCARG(cons)) ENCARG(cons_app(not, filter)) -> c10(APP(not, filter), ENCARG(filter)) ENCARG(cons_app(not, filter2)) -> c10(APP(not, filter2), ENCARG(filter2)) ENCARG(cons_app(not, true)) -> c10(APP(not, true), ENCARG(true)) ENCARG(cons_app(not, false)) -> c10(APP(not, false), ENCARG(false)) ENCARG(cons_app(not, cons_app(z0, z1))) -> c10(APP(not, app(encArg(z0), encArg(z1))), ENCARG(cons_app(z0, z1))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(not, not)) -> c10(APP(not, not), ENCARG(not)) ENCARG(cons_app(not, or)) -> c10(APP(not, or), ENCARG(or)) ENCARG(cons_app(not, and)) -> c10(APP(not, and), ENCARG(and)) ENCARG(cons_app(not, map)) -> c10(APP(not, map), ENCARG(map)) ENCARG(cons_app(not, nil)) -> c10(APP(not, nil), ENCARG(nil)) ENCARG(cons_app(not, cons)) -> c10(APP(not, cons), ENCARG(cons)) ENCARG(cons_app(not, filter)) -> c10(APP(not, filter), ENCARG(filter)) ENCARG(cons_app(not, filter2)) -> c10(APP(not, filter2), ENCARG(filter2)) ENCARG(cons_app(not, true)) -> c10(APP(not, true), ENCARG(true)) ENCARG(cons_app(not, false)) -> c10(APP(not, false), ENCARG(false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(not, and)) -> c10(APP(not, and), ENCARG(and)) ENCARG(cons_app(not, true)) -> c10(APP(not, true), ENCARG(true)) ENCARG(cons_app(not, cons)) -> c10(APP(not, cons), ENCARG(cons)) ENCARG(cons_app(not, nil)) -> c10(APP(not, nil), ENCARG(nil)) ENCARG(cons_app(not, or)) -> c10(APP(not, or), ENCARG(or)) ENCARG(cons_app(not, not)) -> c10(APP(not, not), ENCARG(not)) ENCARG(cons_app(not, filter2)) -> c10(APP(not, filter2), ENCARG(filter2)) ENCARG(cons_app(not, filter)) -> c10(APP(not, filter), ENCARG(filter)) ENCARG(cons_app(not, map)) -> c10(APP(not, map), ENCARG(map)) ENCARG(cons_app(not, false)) -> c10(APP(not, false), ENCARG(false)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(not, app(app(or, x0), app(app(or, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(or, z0), z1))) by APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(not, app(app(or, app(app(or, z0), z1)), x1)) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(or, z0), z1)), APP(not, x1)) by APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(or, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(and, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(or, z0), z1))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2), APP(not, app(app(or, z0), z1))) by APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) by APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(or, app(not, z0)), app(not, z1))), not), app(app(and, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(and, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2))), app(and, z0)), app(app(or, z1), z2)), x2)), app(filter, app(and, z0))), app(app(cons, app(app(or, z1), z2)), x2)), x3), APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1))), app(and, app(app(or, z0), z1))), z2), x2)), app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(map, z0)), nil), x2)), app(filter, app(map, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(map, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(filter, z0)), nil), x2)), app(filter, app(filter, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(or, app(not, z0)), app(not, z1))), not), app(app(and, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(and, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2))), app(and, z0)), app(app(or, z1), z2)), x2)), app(filter, app(and, z0))), app(app(cons, app(app(or, z1), z2)), x2)), x3), APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1))), app(and, app(app(or, z0), z1))), z2), x2)), app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(map, z0)), nil), x2)), app(filter, app(map, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(map, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(filter, z0)), nil), x2)), app(filter, app(filter, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(or, app(not, z0)), app(not, z1))), not), app(app(and, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(and, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2))), app(and, z0)), app(app(or, z1), z2)), x2)), app(filter, app(and, z0))), app(app(cons, app(app(or, z1), z2)), x2)), x3), APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1))), app(and, app(app(or, z0), z1))), z2), x2)), app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(map, z0)), nil), x2)), app(filter, app(map, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(map, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(filter, z0)), nil), x2)), app(filter, app(filter, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) by APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, nil), app(app(app(filter2, false), z0), x1)), nil), x3), APP(app(app(app(filter2, false), z0), x1), nil)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, nil), app(app(app(filter2, false), z0), x1)), nil), x3), APP(app(app(app(filter2, false), z0), x1), nil)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, nil), app(app(app(filter2, false), z0), x1)), nil), x3), APP(app(app(app(filter2, false), z0), x1), nil)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (125) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, not) -> c(APP(encArg(x0), not)) by ENCODE_APP(not, not) -> c(APP(not, not)) ENCODE_APP(or, not) -> c(APP(or, not)) ENCODE_APP(and, not) -> c(APP(and, not)) ENCODE_APP(map, not) -> c(APP(map, not)) ENCODE_APP(nil, not) -> c(APP(nil, not)) ENCODE_APP(cons, not) -> c(APP(cons, not)) ENCODE_APP(filter, not) -> c(APP(filter, not)) ENCODE_APP(filter2, not) -> c(APP(filter2, not)) ENCODE_APP(true, not) -> c(APP(true, not)) ENCODE_APP(false, not) -> c(APP(false, not)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(not, not) -> c(APP(not, not)) ENCODE_APP(or, not) -> c(APP(or, not)) ENCODE_APP(and, not) -> c(APP(and, not)) ENCODE_APP(map, not) -> c(APP(map, not)) ENCODE_APP(nil, not) -> c(APP(nil, not)) ENCODE_APP(cons, not) -> c(APP(cons, not)) ENCODE_APP(filter, not) -> c(APP(filter, not)) ENCODE_APP(filter2, not) -> c(APP(filter2, not)) ENCODE_APP(true, not) -> c(APP(true, not)) ENCODE_APP(false, not) -> c(APP(false, not)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(false, not) -> c(APP(false, not)) ENCODE_APP(filter2, not) -> c(APP(filter2, not)) ENCODE_APP(not, not) -> c(APP(not, not)) ENCODE_APP(true, not) -> c(APP(true, not)) ENCODE_APP(and, not) -> c(APP(and, not)) ENCODE_APP(cons, not) -> c(APP(cons, not)) ENCODE_APP(or, not) -> c(APP(or, not)) ENCODE_APP(filter, not) -> c(APP(filter, not)) ENCODE_APP(nil, not) -> c(APP(nil, not)) ENCODE_APP(map, not) -> c(APP(map, not)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, or) -> c(APP(encArg(x0), or)) by ENCODE_APP(not, or) -> c(APP(not, or)) ENCODE_APP(or, or) -> c(APP(or, or)) ENCODE_APP(and, or) -> c(APP(and, or)) ENCODE_APP(map, or) -> c(APP(map, or)) ENCODE_APP(nil, or) -> c(APP(nil, or)) ENCODE_APP(cons, or) -> c(APP(cons, or)) ENCODE_APP(filter, or) -> c(APP(filter, or)) ENCODE_APP(filter2, or) -> c(APP(filter2, or)) ENCODE_APP(true, or) -> c(APP(true, or)) ENCODE_APP(false, or) -> c(APP(false, or)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(not, or) -> c(APP(not, or)) ENCODE_APP(or, or) -> c(APP(or, or)) ENCODE_APP(and, or) -> c(APP(and, or)) ENCODE_APP(map, or) -> c(APP(map, or)) ENCODE_APP(nil, or) -> c(APP(nil, or)) ENCODE_APP(cons, or) -> c(APP(cons, or)) ENCODE_APP(filter, or) -> c(APP(filter, or)) ENCODE_APP(filter2, or) -> c(APP(filter2, or)) ENCODE_APP(true, or) -> c(APP(true, or)) ENCODE_APP(false, or) -> c(APP(false, or)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(or, or) -> c(APP(or, or)) ENCODE_APP(true, or) -> c(APP(true, or)) ENCODE_APP(nil, or) -> c(APP(nil, or)) ENCODE_APP(map, or) -> c(APP(map, or)) ENCODE_APP(filter2, or) -> c(APP(filter2, or)) ENCODE_APP(false, or) -> c(APP(false, or)) ENCODE_APP(cons, or) -> c(APP(cons, or)) ENCODE_APP(and, or) -> c(APP(and, or)) ENCODE_APP(not, or) -> c(APP(not, or)) ENCODE_APP(filter, or) -> c(APP(filter, or)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, and) -> c(APP(encArg(x0), and)) by ENCODE_APP(not, and) -> c(APP(not, and)) ENCODE_APP(or, and) -> c(APP(or, and)) ENCODE_APP(and, and) -> c(APP(and, and)) ENCODE_APP(map, and) -> c(APP(map, and)) ENCODE_APP(nil, and) -> c(APP(nil, and)) ENCODE_APP(cons, and) -> c(APP(cons, and)) ENCODE_APP(filter, and) -> c(APP(filter, and)) ENCODE_APP(filter2, and) -> c(APP(filter2, and)) ENCODE_APP(true, and) -> c(APP(true, and)) ENCODE_APP(false, and) -> c(APP(false, and)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(not, and) -> c(APP(not, and)) ENCODE_APP(or, and) -> c(APP(or, and)) ENCODE_APP(and, and) -> c(APP(and, and)) ENCODE_APP(map, and) -> c(APP(map, and)) ENCODE_APP(nil, and) -> c(APP(nil, and)) ENCODE_APP(cons, and) -> c(APP(cons, and)) ENCODE_APP(filter, and) -> c(APP(filter, and)) ENCODE_APP(filter2, and) -> c(APP(filter2, and)) ENCODE_APP(true, and) -> c(APP(true, and)) ENCODE_APP(false, and) -> c(APP(false, and)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(cons, and) -> c(APP(cons, and)) ENCODE_APP(filter2, and) -> c(APP(filter2, and)) ENCODE_APP(and, and) -> c(APP(and, and)) ENCODE_APP(true, and) -> c(APP(true, and)) ENCODE_APP(not, and) -> c(APP(not, and)) ENCODE_APP(nil, and) -> c(APP(nil, and)) ENCODE_APP(map, and) -> c(APP(map, and)) ENCODE_APP(filter, and) -> c(APP(filter, and)) ENCODE_APP(false, and) -> c(APP(false, and)) ENCODE_APP(or, and) -> c(APP(or, and)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) by ENCODE_APP(not, map) -> c(APP(not, map)) ENCODE_APP(or, map) -> c(APP(or, map)) ENCODE_APP(and, map) -> c(APP(and, map)) ENCODE_APP(map, map) -> c(APP(map, map)) ENCODE_APP(nil, map) -> c(APP(nil, map)) ENCODE_APP(cons, map) -> c(APP(cons, map)) ENCODE_APP(filter, map) -> c(APP(filter, map)) ENCODE_APP(filter2, map) -> c(APP(filter2, map)) ENCODE_APP(true, map) -> c(APP(true, map)) ENCODE_APP(false, map) -> c(APP(false, map)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(not, map) -> c(APP(not, map)) ENCODE_APP(or, map) -> c(APP(or, map)) ENCODE_APP(and, map) -> c(APP(and, map)) ENCODE_APP(map, map) -> c(APP(map, map)) ENCODE_APP(nil, map) -> c(APP(nil, map)) ENCODE_APP(cons, map) -> c(APP(cons, map)) ENCODE_APP(filter, map) -> c(APP(filter, map)) ENCODE_APP(filter2, map) -> c(APP(filter2, map)) ENCODE_APP(true, map) -> c(APP(true, map)) ENCODE_APP(false, map) -> c(APP(false, map)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(filter2, map) -> c(APP(filter2, map)) ENCODE_APP(filter, map) -> c(APP(filter, map)) ENCODE_APP(map, map) -> c(APP(map, map)) ENCODE_APP(cons, map) -> c(APP(cons, map)) ENCODE_APP(false, map) -> c(APP(false, map)) ENCODE_APP(true, map) -> c(APP(true, map)) ENCODE_APP(nil, map) -> c(APP(nil, map)) ENCODE_APP(or, map) -> c(APP(or, map)) ENCODE_APP(not, map) -> c(APP(not, map)) ENCODE_APP(and, map) -> c(APP(and, map)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) by ENCODE_APP(not, nil) -> c(APP(not, nil)) ENCODE_APP(or, nil) -> c(APP(or, nil)) ENCODE_APP(and, nil) -> c(APP(and, nil)) ENCODE_APP(map, nil) -> c(APP(map, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(filter2, nil) -> c(APP(filter2, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(not, nil) -> c(APP(not, nil)) ENCODE_APP(or, nil) -> c(APP(or, nil)) ENCODE_APP(and, nil) -> c(APP(and, nil)) ENCODE_APP(map, nil) -> c(APP(map, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(filter2, nil) -> c(APP(filter2, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(not, nil) -> c(APP(not, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(and, nil) -> c(APP(and, nil)) ENCODE_APP(map, nil) -> c(APP(map, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ENCODE_APP(or, nil) -> c(APP(or, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(filter2, nil) -> c(APP(filter2, nil)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) by ENCODE_APP(not, cons) -> c(APP(not, cons)) ENCODE_APP(or, cons) -> c(APP(or, cons)) ENCODE_APP(and, cons) -> c(APP(and, cons)) ENCODE_APP(map, cons) -> c(APP(map, cons)) ENCODE_APP(nil, cons) -> c(APP(nil, cons)) ENCODE_APP(cons, cons) -> c(APP(cons, cons)) ENCODE_APP(filter, cons) -> c(APP(filter, cons)) ENCODE_APP(filter2, cons) -> c(APP(filter2, cons)) ENCODE_APP(true, cons) -> c(APP(true, cons)) ENCODE_APP(false, cons) -> c(APP(false, cons)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(not, cons) -> c(APP(not, cons)) ENCODE_APP(or, cons) -> c(APP(or, cons)) ENCODE_APP(and, cons) -> c(APP(and, cons)) ENCODE_APP(map, cons) -> c(APP(map, cons)) ENCODE_APP(nil, cons) -> c(APP(nil, cons)) ENCODE_APP(cons, cons) -> c(APP(cons, cons)) ENCODE_APP(filter, cons) -> c(APP(filter, cons)) ENCODE_APP(filter2, cons) -> c(APP(filter2, cons)) ENCODE_APP(true, cons) -> c(APP(true, cons)) ENCODE_APP(false, cons) -> c(APP(false, cons)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (149) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(cons, cons) -> c(APP(cons, cons)) ENCODE_APP(nil, cons) -> c(APP(nil, cons)) ENCODE_APP(not, cons) -> c(APP(not, cons)) ENCODE_APP(filter2, cons) -> c(APP(filter2, cons)) ENCODE_APP(filter, cons) -> c(APP(filter, cons)) ENCODE_APP(or, cons) -> c(APP(or, cons)) ENCODE_APP(false, cons) -> c(APP(false, cons)) ENCODE_APP(and, cons) -> c(APP(and, cons)) ENCODE_APP(true, cons) -> c(APP(true, cons)) ENCODE_APP(map, cons) -> c(APP(map, cons)) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) by ENCODE_APP(not, filter) -> c(APP(not, filter)) ENCODE_APP(or, filter) -> c(APP(or, filter)) ENCODE_APP(and, filter) -> c(APP(and, filter)) ENCODE_APP(map, filter) -> c(APP(map, filter)) ENCODE_APP(nil, filter) -> c(APP(nil, filter)) ENCODE_APP(cons, filter) -> c(APP(cons, filter)) ENCODE_APP(filter, filter) -> c(APP(filter, filter)) ENCODE_APP(filter2, filter) -> c(APP(filter2, filter)) ENCODE_APP(true, filter) -> c(APP(true, filter)) ENCODE_APP(false, filter) -> c(APP(false, filter)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(not, filter) -> c(APP(not, filter)) ENCODE_APP(or, filter) -> c(APP(or, filter)) ENCODE_APP(and, filter) -> c(APP(and, filter)) ENCODE_APP(map, filter) -> c(APP(map, filter)) ENCODE_APP(nil, filter) -> c(APP(nil, filter)) ENCODE_APP(cons, filter) -> c(APP(cons, filter)) ENCODE_APP(filter, filter) -> c(APP(filter, filter)) ENCODE_APP(filter2, filter) -> c(APP(filter2, filter)) ENCODE_APP(true, filter) -> c(APP(true, filter)) ENCODE_APP(false, filter) -> c(APP(false, filter)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (153) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(filter2, filter) -> c(APP(filter2, filter)) ENCODE_APP(filter, filter) -> c(APP(filter, filter)) ENCODE_APP(not, filter) -> c(APP(not, filter)) ENCODE_APP(and, filter) -> c(APP(and, filter)) ENCODE_APP(map, filter) -> c(APP(map, filter)) ENCODE_APP(true, filter) -> c(APP(true, filter)) ENCODE_APP(cons, filter) -> c(APP(cons, filter)) ENCODE_APP(nil, filter) -> c(APP(nil, filter)) ENCODE_APP(false, filter) -> c(APP(false, filter)) ENCODE_APP(or, filter) -> c(APP(or, filter)) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) by ENCODE_APP(not, filter2) -> c(APP(not, filter2)) ENCODE_APP(or, filter2) -> c(APP(or, filter2)) ENCODE_APP(and, filter2) -> c(APP(and, filter2)) ENCODE_APP(map, filter2) -> c(APP(map, filter2)) ENCODE_APP(nil, filter2) -> c(APP(nil, filter2)) ENCODE_APP(cons, filter2) -> c(APP(cons, filter2)) ENCODE_APP(filter, filter2) -> c(APP(filter, filter2)) ENCODE_APP(filter2, filter2) -> c(APP(filter2, filter2)) ENCODE_APP(true, filter2) -> c(APP(true, filter2)) ENCODE_APP(false, filter2) -> c(APP(false, filter2)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(not, filter2) -> c(APP(not, filter2)) ENCODE_APP(or, filter2) -> c(APP(or, filter2)) ENCODE_APP(and, filter2) -> c(APP(and, filter2)) ENCODE_APP(map, filter2) -> c(APP(map, filter2)) ENCODE_APP(nil, filter2) -> c(APP(nil, filter2)) ENCODE_APP(cons, filter2) -> c(APP(cons, filter2)) ENCODE_APP(filter, filter2) -> c(APP(filter, filter2)) ENCODE_APP(filter2, filter2) -> c(APP(filter2, filter2)) ENCODE_APP(true, filter2) -> c(APP(true, filter2)) ENCODE_APP(false, filter2) -> c(APP(false, filter2)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(true, filter2) -> c(APP(true, filter2)) ENCODE_APP(filter2, filter2) -> c(APP(filter2, filter2)) ENCODE_APP(and, filter2) -> c(APP(and, filter2)) ENCODE_APP(not, filter2) -> c(APP(not, filter2)) ENCODE_APP(filter, filter2) -> c(APP(filter, filter2)) ENCODE_APP(or, filter2) -> c(APP(or, filter2)) ENCODE_APP(false, filter2) -> c(APP(false, filter2)) ENCODE_APP(map, filter2) -> c(APP(map, filter2)) ENCODE_APP(nil, filter2) -> c(APP(nil, filter2)) ENCODE_APP(cons, filter2) -> c(APP(cons, filter2)) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) by ENCODE_APP(not, true) -> c(APP(not, true)) ENCODE_APP(or, true) -> c(APP(or, true)) ENCODE_APP(and, true) -> c(APP(and, true)) ENCODE_APP(map, true) -> c(APP(map, true)) ENCODE_APP(nil, true) -> c(APP(nil, true)) ENCODE_APP(cons, true) -> c(APP(cons, true)) ENCODE_APP(filter, true) -> c(APP(filter, true)) ENCODE_APP(filter2, true) -> c(APP(filter2, true)) ENCODE_APP(true, true) -> c(APP(true, true)) ENCODE_APP(false, true) -> c(APP(false, true)) ENCODE_APP(cons_app(z0, z1), true) -> c(APP(app(encArg(z0), encArg(z1)), true)) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) ENCODE_APP(not, true) -> c(APP(not, true)) ENCODE_APP(or, true) -> c(APP(or, true)) ENCODE_APP(and, true) -> c(APP(and, true)) ENCODE_APP(map, true) -> c(APP(map, true)) ENCODE_APP(nil, true) -> c(APP(nil, true)) ENCODE_APP(cons, true) -> c(APP(cons, true)) ENCODE_APP(filter, true) -> c(APP(filter, true)) ENCODE_APP(filter2, true) -> c(APP(filter2, true)) ENCODE_APP(true, true) -> c(APP(true, true)) ENCODE_APP(false, true) -> c(APP(false, true)) ENCODE_APP(cons_app(z0, z1), true) -> c(APP(app(encArg(z0), encArg(z1)), true)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(false, true) -> c(APP(false, true)) ENCODE_APP(filter, true) -> c(APP(filter, true)) ENCODE_APP(or, true) -> c(APP(or, true)) ENCODE_APP(filter2, true) -> c(APP(filter2, true)) ENCODE_APP(map, true) -> c(APP(map, true)) ENCODE_APP(and, true) -> c(APP(and, true)) ENCODE_APP(not, true) -> c(APP(not, true)) ENCODE_APP(cons, true) -> c(APP(cons, true)) ENCODE_APP(nil, true) -> c(APP(nil, true)) ENCODE_APP(true, true) -> c(APP(true, true)) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) ENCODE_APP(cons_app(z0, z1), true) -> c(APP(app(encArg(z0), encArg(z1)), true)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) by ENCODE_APP(not, false) -> c(APP(not, false)) ENCODE_APP(or, false) -> c(APP(or, false)) ENCODE_APP(and, false) -> c(APP(and, false)) ENCODE_APP(map, false) -> c(APP(map, false)) ENCODE_APP(nil, false) -> c(APP(nil, false)) ENCODE_APP(cons, false) -> c(APP(cons, false)) ENCODE_APP(filter, false) -> c(APP(filter, false)) ENCODE_APP(filter2, false) -> c(APP(filter2, false)) ENCODE_APP(true, false) -> c(APP(true, false)) ENCODE_APP(false, false) -> c(APP(false, false)) ENCODE_APP(cons_app(z0, z1), false) -> c(APP(app(encArg(z0), encArg(z1)), false)) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) ENCODE_APP(cons_app(z0, z1), true) -> c(APP(app(encArg(z0), encArg(z1)), true)) ENCODE_APP(not, false) -> c(APP(not, false)) ENCODE_APP(or, false) -> c(APP(or, false)) ENCODE_APP(and, false) -> c(APP(and, false)) ENCODE_APP(map, false) -> c(APP(map, false)) ENCODE_APP(nil, false) -> c(APP(nil, false)) ENCODE_APP(cons, false) -> c(APP(cons, false)) ENCODE_APP(filter, false) -> c(APP(filter, false)) ENCODE_APP(filter2, false) -> c(APP(filter2, false)) ENCODE_APP(true, false) -> c(APP(true, false)) ENCODE_APP(false, false) -> c(APP(false, false)) ENCODE_APP(cons_app(z0, z1), false) -> c(APP(app(encArg(z0), encArg(z1)), false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (165) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(filter, false) -> c(APP(filter, false)) ENCODE_APP(cons, false) -> c(APP(cons, false)) ENCODE_APP(and, false) -> c(APP(and, false)) ENCODE_APP(true, false) -> c(APP(true, false)) ENCODE_APP(false, false) -> c(APP(false, false)) ENCODE_APP(or, false) -> c(APP(or, false)) ENCODE_APP(not, false) -> c(APP(not, false)) ENCODE_APP(filter2, false) -> c(APP(filter2, false)) ENCODE_APP(nil, false) -> c(APP(nil, false)) ENCODE_APP(map, false) -> c(APP(map, false)) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) ENCODE_APP(cons_app(z0, z1), true) -> c(APP(app(encArg(z0), encArg(z1)), true)) ENCODE_APP(cons_app(z0, z1), false) -> c(APP(app(encArg(z0), encArg(z1)), false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (167) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) by ENCODE_APP(x0, cons_app(x1, not)) -> c(APP(encArg(x0), app(encArg(x1), not))) ENCODE_APP(x0, cons_app(x1, or)) -> c(APP(encArg(x0), app(encArg(x1), or))) ENCODE_APP(x0, cons_app(x1, and)) -> c(APP(encArg(x0), app(encArg(x1), and))) ENCODE_APP(x0, cons_app(x1, map)) -> c(APP(encArg(x0), app(encArg(x1), map))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, filter2)) -> c(APP(encArg(x0), app(encArg(x1), filter2))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(not, x2)) -> c(APP(encArg(x0), app(not, encArg(x2)))) ENCODE_APP(x0, cons_app(or, x2)) -> c(APP(encArg(x0), app(or, encArg(x2)))) ENCODE_APP(x0, cons_app(and, x2)) -> c(APP(encArg(x0), app(and, encArg(x2)))) ENCODE_APP(x0, cons_app(map, x2)) -> c(APP(encArg(x0), app(map, encArg(x2)))) ENCODE_APP(x0, cons_app(nil, x2)) -> c(APP(encArg(x0), app(nil, encArg(x2)))) ENCODE_APP(x0, cons_app(cons, x2)) -> c(APP(encArg(x0), app(cons, encArg(x2)))) ENCODE_APP(x0, cons_app(filter, x2)) -> c(APP(encArg(x0), app(filter, encArg(x2)))) ENCODE_APP(x0, cons_app(filter2, x2)) -> c(APP(encArg(x0), app(filter2, encArg(x2)))) ENCODE_APP(x0, cons_app(true, x2)) -> c(APP(encArg(x0), app(true, encArg(x2)))) ENCODE_APP(x0, cons_app(false, x2)) -> c(APP(encArg(x0), app(false, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(not, cons_app(x1, x2)) -> c(APP(not, app(encArg(x1), encArg(x2)))) ENCODE_APP(or, cons_app(x1, x2)) -> c(APP(or, app(encArg(x1), encArg(x2)))) ENCODE_APP(and, cons_app(x1, x2)) -> c(APP(and, app(encArg(x1), encArg(x2)))) ENCODE_APP(map, cons_app(x1, x2)) -> c(APP(map, app(encArg(x1), encArg(x2)))) ENCODE_APP(nil, cons_app(x1, x2)) -> c(APP(nil, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons, cons_app(x1, x2)) -> c(APP(cons, app(encArg(x1), encArg(x2)))) ENCODE_APP(filter, cons_app(x1, x2)) -> c(APP(filter, app(encArg(x1), encArg(x2)))) ENCODE_APP(filter2, cons_app(x1, x2)) -> c(APP(filter2, app(encArg(x1), encArg(x2)))) ENCODE_APP(true, cons_app(x1, x2)) -> c(APP(true, app(encArg(x1), encArg(x2)))) ENCODE_APP(false, cons_app(x1, x2)) -> c(APP(false, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) ENCODE_APP(cons_app(z0, z1), true) -> c(APP(app(encArg(z0), encArg(z1)), true)) ENCODE_APP(cons_app(z0, z1), false) -> c(APP(app(encArg(z0), encArg(z1)), false)) ENCODE_APP(x0, cons_app(x1, not)) -> c(APP(encArg(x0), app(encArg(x1), not))) ENCODE_APP(x0, cons_app(x1, or)) -> c(APP(encArg(x0), app(encArg(x1), or))) ENCODE_APP(x0, cons_app(x1, and)) -> c(APP(encArg(x0), app(encArg(x1), and))) ENCODE_APP(x0, cons_app(x1, map)) -> c(APP(encArg(x0), app(encArg(x1), map))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, filter2)) -> c(APP(encArg(x0), app(encArg(x1), filter2))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(not, x2)) -> c(APP(encArg(x0), app(not, encArg(x2)))) ENCODE_APP(x0, cons_app(or, x2)) -> c(APP(encArg(x0), app(or, encArg(x2)))) ENCODE_APP(x0, cons_app(and, x2)) -> c(APP(encArg(x0), app(and, encArg(x2)))) ENCODE_APP(x0, cons_app(map, x2)) -> c(APP(encArg(x0), app(map, encArg(x2)))) ENCODE_APP(x0, cons_app(nil, x2)) -> c(APP(encArg(x0), app(nil, encArg(x2)))) ENCODE_APP(x0, cons_app(cons, x2)) -> c(APP(encArg(x0), app(cons, encArg(x2)))) ENCODE_APP(x0, cons_app(filter, x2)) -> c(APP(encArg(x0), app(filter, encArg(x2)))) ENCODE_APP(x0, cons_app(filter2, x2)) -> c(APP(encArg(x0), app(filter2, encArg(x2)))) ENCODE_APP(x0, cons_app(true, x2)) -> c(APP(encArg(x0), app(true, encArg(x2)))) ENCODE_APP(x0, cons_app(false, x2)) -> c(APP(encArg(x0), app(false, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(not, cons_app(x1, x2)) -> c(APP(not, app(encArg(x1), encArg(x2)))) ENCODE_APP(or, cons_app(x1, x2)) -> c(APP(or, app(encArg(x1), encArg(x2)))) ENCODE_APP(and, cons_app(x1, x2)) -> c(APP(and, app(encArg(x1), encArg(x2)))) ENCODE_APP(map, cons_app(x1, x2)) -> c(APP(map, app(encArg(x1), encArg(x2)))) ENCODE_APP(nil, cons_app(x1, x2)) -> c(APP(nil, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons, cons_app(x1, x2)) -> c(APP(cons, app(encArg(x1), encArg(x2)))) ENCODE_APP(filter, cons_app(x1, x2)) -> c(APP(filter, app(encArg(x1), encArg(x2)))) ENCODE_APP(filter2, cons_app(x1, x2)) -> c(APP(filter2, app(encArg(x1), encArg(x2)))) ENCODE_APP(true, cons_app(x1, x2)) -> c(APP(true, app(encArg(x1), encArg(x2)))) ENCODE_APP(false, cons_app(x1, x2)) -> c(APP(false, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(nil, cons_app(x1, x2)) -> c(APP(nil, app(encArg(x1), encArg(x2)))) ENCODE_APP(false, cons_app(x1, x2)) -> c(APP(false, app(encArg(x1), encArg(x2)))) ENCODE_APP(or, cons_app(x1, x2)) -> c(APP(or, app(encArg(x1), encArg(x2)))) ENCODE_APP(true, cons_app(x1, x2)) -> c(APP(true, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons, cons_app(x1, x2)) -> c(APP(cons, app(encArg(x1), encArg(x2)))) ENCODE_APP(map, cons_app(x1, x2)) -> c(APP(map, app(encArg(x1), encArg(x2)))) ENCODE_APP(and, cons_app(x1, x2)) -> c(APP(and, app(encArg(x1), encArg(x2)))) ENCODE_APP(filter2, cons_app(x1, x2)) -> c(APP(filter2, app(encArg(x1), encArg(x2)))) ENCODE_APP(filter, cons_app(x1, x2)) -> c(APP(filter, app(encArg(x1), encArg(x2)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) ENCODE_APP(cons_app(z0, z1), true) -> c(APP(app(encArg(z0), encArg(z1)), true)) ENCODE_APP(cons_app(z0, z1), false) -> c(APP(app(encArg(z0), encArg(z1)), false)) ENCODE_APP(x0, cons_app(x1, not)) -> c(APP(encArg(x0), app(encArg(x1), not))) ENCODE_APP(x0, cons_app(x1, or)) -> c(APP(encArg(x0), app(encArg(x1), or))) ENCODE_APP(x0, cons_app(x1, and)) -> c(APP(encArg(x0), app(encArg(x1), and))) ENCODE_APP(x0, cons_app(x1, map)) -> c(APP(encArg(x0), app(encArg(x1), map))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, filter2)) -> c(APP(encArg(x0), app(encArg(x1), filter2))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(not, x2)) -> c(APP(encArg(x0), app(not, encArg(x2)))) ENCODE_APP(x0, cons_app(or, x2)) -> c(APP(encArg(x0), app(or, encArg(x2)))) ENCODE_APP(x0, cons_app(and, x2)) -> c(APP(encArg(x0), app(and, encArg(x2)))) ENCODE_APP(x0, cons_app(map, x2)) -> c(APP(encArg(x0), app(map, encArg(x2)))) ENCODE_APP(x0, cons_app(nil, x2)) -> c(APP(encArg(x0), app(nil, encArg(x2)))) ENCODE_APP(x0, cons_app(cons, x2)) -> c(APP(encArg(x0), app(cons, encArg(x2)))) ENCODE_APP(x0, cons_app(filter, x2)) -> c(APP(encArg(x0), app(filter, encArg(x2)))) ENCODE_APP(x0, cons_app(filter2, x2)) -> c(APP(encArg(x0), app(filter2, encArg(x2)))) ENCODE_APP(x0, cons_app(true, x2)) -> c(APP(encArg(x0), app(true, encArg(x2)))) ENCODE_APP(x0, cons_app(false, x2)) -> c(APP(encArg(x0), app(false, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(not, cons_app(x1, x2)) -> c(APP(not, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (171) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(not, x1) -> c(APP(not, encArg(x1))) by ENCODE_APP(not, not) -> c(APP(not, not)) ENCODE_APP(not, or) -> c(APP(not, or)) ENCODE_APP(not, and) -> c(APP(not, and)) ENCODE_APP(not, map) -> c(APP(not, map)) ENCODE_APP(not, nil) -> c(APP(not, nil)) ENCODE_APP(not, cons) -> c(APP(not, cons)) ENCODE_APP(not, filter) -> c(APP(not, filter)) ENCODE_APP(not, filter2) -> c(APP(not, filter2)) ENCODE_APP(not, true) -> c(APP(not, true)) ENCODE_APP(not, false) -> c(APP(not, false)) ENCODE_APP(not, cons_app(z0, z1)) -> c(APP(not, app(encArg(z0), encArg(z1)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) ENCODE_APP(cons_app(z0, z1), true) -> c(APP(app(encArg(z0), encArg(z1)), true)) ENCODE_APP(cons_app(z0, z1), false) -> c(APP(app(encArg(z0), encArg(z1)), false)) ENCODE_APP(x0, cons_app(x1, not)) -> c(APP(encArg(x0), app(encArg(x1), not))) ENCODE_APP(x0, cons_app(x1, or)) -> c(APP(encArg(x0), app(encArg(x1), or))) ENCODE_APP(x0, cons_app(x1, and)) -> c(APP(encArg(x0), app(encArg(x1), and))) ENCODE_APP(x0, cons_app(x1, map)) -> c(APP(encArg(x0), app(encArg(x1), map))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, filter2)) -> c(APP(encArg(x0), app(encArg(x1), filter2))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(not, x2)) -> c(APP(encArg(x0), app(not, encArg(x2)))) ENCODE_APP(x0, cons_app(or, x2)) -> c(APP(encArg(x0), app(or, encArg(x2)))) ENCODE_APP(x0, cons_app(and, x2)) -> c(APP(encArg(x0), app(and, encArg(x2)))) ENCODE_APP(x0, cons_app(map, x2)) -> c(APP(encArg(x0), app(map, encArg(x2)))) ENCODE_APP(x0, cons_app(nil, x2)) -> c(APP(encArg(x0), app(nil, encArg(x2)))) ENCODE_APP(x0, cons_app(cons, x2)) -> c(APP(encArg(x0), app(cons, encArg(x2)))) ENCODE_APP(x0, cons_app(filter, x2)) -> c(APP(encArg(x0), app(filter, encArg(x2)))) ENCODE_APP(x0, cons_app(filter2, x2)) -> c(APP(encArg(x0), app(filter2, encArg(x2)))) ENCODE_APP(x0, cons_app(true, x2)) -> c(APP(encArg(x0), app(true, encArg(x2)))) ENCODE_APP(x0, cons_app(false, x2)) -> c(APP(encArg(x0), app(false, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(not, cons_app(x1, x2)) -> c(APP(not, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(not, not) -> c(APP(not, not)) ENCODE_APP(not, or) -> c(APP(not, or)) ENCODE_APP(not, and) -> c(APP(not, and)) ENCODE_APP(not, map) -> c(APP(not, map)) ENCODE_APP(not, nil) -> c(APP(not, nil)) ENCODE_APP(not, cons) -> c(APP(not, cons)) ENCODE_APP(not, filter) -> c(APP(not, filter)) ENCODE_APP(not, filter2) -> c(APP(not, filter2)) ENCODE_APP(not, true) -> c(APP(not, true)) ENCODE_APP(not, false) -> c(APP(not, false)) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2 ---------------------------------------- (173) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(not, not) -> c(APP(not, not)) ENCODE_APP(not, filter2) -> c(APP(not, filter2)) ENCODE_APP(not, cons) -> c(APP(not, cons)) ENCODE_APP(not, and) -> c(APP(not, and)) ENCODE_APP(not, map) -> c(APP(not, map)) ENCODE_APP(not, filter) -> c(APP(not, filter)) ENCODE_APP(not, false) -> c(APP(not, false)) ENCODE_APP(not, true) -> c(APP(not, true)) ENCODE_APP(not, or) -> c(APP(not, or)) ENCODE_APP(not, nil) -> c(APP(not, nil)) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: encArg(not) -> not encArg(or) -> or encArg(and) -> and encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(not, app(not, z0)) -> z0 app(not, app(app(or, z0), z1)) -> app(app(and, app(not, z0)), app(not, z1)) app(not, app(app(and, z0), z1)) -> app(app(or, app(not, z0)), app(not, z1)) app(app(and, z0), app(app(or, z1), z2)) -> app(app(or, app(app(and, z0), z1)), app(app(and, z0), z2)) app(app(and, app(app(or, z0), z1)), z2) -> app(app(or, app(app(and, z2), z0)), app(app(and, z2), z1)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(or, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(and, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, not))) -> c10(APP(encArg(x0), app(encArg(x1), not)), ENCARG(x0), ENCARG(cons_app(x1, not))) ENCARG(cons_app(x0, cons_app(x1, or))) -> c10(APP(encArg(x0), app(encArg(x1), or)), ENCARG(x0), ENCARG(cons_app(x1, or))) ENCARG(cons_app(x0, cons_app(x1, and))) -> c10(APP(encArg(x0), app(encArg(x1), and)), ENCARG(x0), ENCARG(cons_app(x1, and))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(not, x2))) -> c10(APP(encArg(x0), app(not, encArg(x2))), ENCARG(x0), ENCARG(cons_app(not, x2))) ENCARG(cons_app(x0, cons_app(or, x2))) -> c10(APP(encArg(x0), app(or, encArg(x2))), ENCARG(x0), ENCARG(cons_app(or, x2))) ENCARG(cons_app(x0, cons_app(and, x2))) -> c10(APP(encArg(x0), app(and, encArg(x2))), ENCARG(x0), ENCARG(cons_app(and, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(not, cons_app(x1, x2))) -> c10(APP(not, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(or, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(and, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, not), x2)) -> c10(APP(app(encArg(x0), not), encArg(x2)), ENCARG(cons_app(x0, not)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, or), x2)) -> c10(APP(app(encArg(x0), or), encArg(x2)), ENCARG(cons_app(x0, or)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, and), x2)) -> c10(APP(app(encArg(x0), and), encArg(x2)), ENCARG(cons_app(x0, and)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(not, x1), x2)) -> c10(APP(app(not, encArg(x1)), encArg(x2)), ENCARG(cons_app(not, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(and, x1), x2)) -> c10(APP(app(and, encArg(x1)), encArg(x2)), ENCARG(cons_app(and, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), not)) -> c10(APP(app(encArg(x0), encArg(x1)), not), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), or)) -> c10(APP(app(encArg(x0), encArg(x1)), or), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), and)) -> c10(APP(app(encArg(x0), encArg(x1)), and), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(or, x1), x2)) -> c10(ENCARG(cons_app(or, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), not) -> c(APP(app(encArg(z0), encArg(z1)), not)) ENCODE_APP(cons_app(z0, z1), or) -> c(APP(app(encArg(z0), encArg(z1)), or)) ENCODE_APP(cons_app(z0, z1), and) -> c(APP(app(encArg(z0), encArg(z1)), and)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ENCODE_APP(cons_app(z0, z1), filter2) -> c(APP(app(encArg(z0), encArg(z1)), filter2)) ENCODE_APP(cons_app(z0, z1), true) -> c(APP(app(encArg(z0), encArg(z1)), true)) ENCODE_APP(cons_app(z0, z1), false) -> c(APP(app(encArg(z0), encArg(z1)), false)) ENCODE_APP(x0, cons_app(x1, not)) -> c(APP(encArg(x0), app(encArg(x1), not))) ENCODE_APP(x0, cons_app(x1, or)) -> c(APP(encArg(x0), app(encArg(x1), or))) ENCODE_APP(x0, cons_app(x1, and)) -> c(APP(encArg(x0), app(encArg(x1), and))) ENCODE_APP(x0, cons_app(x1, map)) -> c(APP(encArg(x0), app(encArg(x1), map))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, filter2)) -> c(APP(encArg(x0), app(encArg(x1), filter2))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(not, x2)) -> c(APP(encArg(x0), app(not, encArg(x2)))) ENCODE_APP(x0, cons_app(or, x2)) -> c(APP(encArg(x0), app(or, encArg(x2)))) ENCODE_APP(x0, cons_app(and, x2)) -> c(APP(encArg(x0), app(and, encArg(x2)))) ENCODE_APP(x0, cons_app(map, x2)) -> c(APP(encArg(x0), app(map, encArg(x2)))) ENCODE_APP(x0, cons_app(nil, x2)) -> c(APP(encArg(x0), app(nil, encArg(x2)))) ENCODE_APP(x0, cons_app(cons, x2)) -> c(APP(encArg(x0), app(cons, encArg(x2)))) ENCODE_APP(x0, cons_app(filter, x2)) -> c(APP(encArg(x0), app(filter, encArg(x2)))) ENCODE_APP(x0, cons_app(filter2, x2)) -> c(APP(encArg(x0), app(filter2, encArg(x2)))) ENCODE_APP(x0, cons_app(true, x2)) -> c(APP(encArg(x0), app(true, encArg(x2)))) ENCODE_APP(x0, cons_app(false, x2)) -> c(APP(encArg(x0), app(false, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(not, cons_app(x1, x2)) -> c(APP(not, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) S tuples: APP(not, app(not, z0)) -> c22 APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(not, app(app(and, z0), z1)) -> c24(APP(not, z0), APP(not, z1)) APP(app(and, z0), app(app(or, z1), z2)) -> c25(APP(app(and, z0), z1), APP(app(and, z0), z2)) APP(app(and, app(app(or, z0), z1)), z2) -> c26(APP(app(and, z2), z0), APP(app(and, z2), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(not, app(app(or, x0), app(not, z0))) -> c23(APP(app(and, app(not, x0)), z0), APP(not, x0), APP(not, app(not, z0))) APP(not, app(app(or, x0), app(app(and, z0), z1))) -> c23(APP(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1))), APP(not, x0), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(not, z0)), x1)) -> c23(APP(app(and, z0), app(not, x1)), APP(not, app(not, z0)), APP(not, x1)) APP(not, app(app(or, app(app(and, z0), z1)), x1)) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1)), APP(not, app(app(and, z0), z1)), APP(not, x1)) APP(app(filter, not), app(app(cons, app(not, z0)), x2)) -> c30(APP(app(app(app(filter2, z0), not), app(not, z0)), x2), APP(not, app(not, z0))) APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2)) -> c30(APP(not, app(app(and, z0), z1))) APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2)) -> c30(APP(app(and, z0), app(app(or, z1), z2))) APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(and, app(app(or, z0), z1)), z2)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(not, app(app(or, x0), app(app(or, x1), app(not, z0)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), z0)), APP(not, x0), APP(not, app(app(or, x1), app(not, z0)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(or, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(and, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(or, z0), z1)))) APP(not, app(app(or, x0), app(app(or, x1), app(app(and, z0), z1)))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(not, x1)), app(app(or, app(not, z0)), app(not, z1)))), APP(not, x0), APP(not, app(app(or, x1), app(app(and, z0), z1)))) APP(not, app(app(or, x0), app(app(or, app(not, z0)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, z0), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(not, z0)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(or, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(or, z0), z1)), x2))) APP(not, app(app(or, x0), app(app(or, app(app(and, z0), z1)), x2))) -> c23(APP(app(and, app(not, x0)), app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x2))), APP(not, x0), APP(not, app(app(or, app(app(and, z0), z1)), x2))) APP(not, app(app(or, app(not, z0)), app(app(or, x1), x2))) -> c23(APP(app(and, z0), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(not, z0)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(and, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(or, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(and, z0), z1)), app(app(or, x1), x2))) -> c23(APP(app(and, app(app(or, app(not, z0)), app(not, z1))), app(app(and, app(not, x1)), app(not, x2))), APP(not, app(app(and, z0), z1)), APP(not, app(app(or, x1), x2))) APP(not, app(app(or, app(app(or, x0), x1)), app(not, z0))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), z0), APP(not, app(app(or, x0), x1)), APP(not, app(not, z0))) APP(not, app(app(or, app(app(or, x0), x1)), app(app(and, z0), z1))) -> c23(APP(app(and, app(app(and, app(not, x0)), app(not, x1))), app(app(or, app(not, z0)), app(not, z1))), APP(not, app(app(or, x0), x1)), APP(not, app(app(and, z0), z1))) APP(not, app(app(or, app(app(or, x0), app(not, z0))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), z0)), app(not, x2)), APP(not, app(app(or, x0), app(not, z0))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(or, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c23(APP(app(and, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), app(not, x2)), APP(not, app(app(or, x0), app(app(and, z0), z1))), APP(not, x2)) APP(not, app(app(or, app(app(or, app(not, z0)), x1)), x2)) -> c23(APP(app(and, app(app(and, z0), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(not, z0)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(or, z0), z1)), x1)), APP(not, x2)) APP(not, app(app(or, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c23(APP(app(and, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), app(not, x2)), APP(not, app(app(or, app(app(and, z0), z1)), x1)), APP(not, x2)) APP(app(filter, not), app(app(cons, app(app(or, x0), app(not, z0))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), z0)), not), app(app(or, x0), app(not, z0))), x2), APP(not, app(app(or, x0), app(not, z0)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(or, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(and, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(or, z0), z1))), x2), APP(not, app(app(or, x0), app(app(or, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, x0), app(app(and, z0), z1))), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(not, x0)), app(app(or, app(not, z0)), app(not, z1)))), not), app(app(or, x0), app(app(and, z0), z1))), x2), APP(not, app(app(or, x0), app(app(and, z0), z1)))) APP(app(filter, not), app(app(cons, app(app(or, app(not, z0)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, z0), app(not, x1))), not), app(app(or, app(not, z0)), x1)), x2), APP(not, app(app(or, app(not, z0)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(or, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(and, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(or, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(or, z0), z1)), x1))) APP(app(filter, not), app(app(cons, app(app(or, app(app(and, z0), z1)), x1)), x2)) -> c30(APP(app(app(app(filter2, app(app(and, app(app(or, app(not, z0)), app(not, z1))), app(not, x1))), not), app(app(or, app(app(and, z0), z1)), x1)), x2), APP(not, app(app(or, app(app(and, z0), z1)), x1))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(not, z0)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, z0), not), app(not, z0)), x2)), app(filter, not)), app(app(cons, app(not, z0)), x2)), x3), APP(app(filter, not), app(app(cons, app(not, z0)), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(or, z0), z1)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(and, app(not, z0)), app(not, z1))), not), app(app(or, z0), z1)), x2)), app(filter, not)), app(app(cons, app(app(or, z0), z1)), x2)), x3), APP(app(filter, not), app(app(cons, app(app(or, z0), z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, not)), app(app(cons, app(app(cons, app(app(and, z0), z1)), x2)), x3)) -> c30(APP(app(filter, not), app(app(cons, app(app(and, z0), z1)), x2))) APP(app(filter, app(filter, app(and, z0))), app(app(cons, app(app(cons, app(app(or, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(and, z0)), app(app(cons, app(app(or, z1), z2)), x2))) APP(app(filter, app(filter, app(and, app(app(or, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(and, app(app(or, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c27, c29, c24_2, c25_2, c26_2, c28_2, c31_1, c32_1, c10_1, c23_3, c30_2, c30_1, c_1, c10_3, c10_2