/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 228 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 8 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 33.0 s] (12) BOUNDS(1, INF) (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (14) TRS for Loop Detection (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), 11 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 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), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 74 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 93 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 4 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 31 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 31 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), 23 ms] (110) CdtProblem (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 18 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 16 ms] (118) CdtProblem (119) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 27 ms] (122) CdtProblem (123) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 16 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (164) 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(pred, app(s, x)) -> x app(app(minus, x), 0) -> x app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y)) app(app(quot, 0), app(s, y)) -> 0 app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot 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(pred, app(s, x)) -> x app(app(minus, x), 0) -> x app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y)) app(app(quot, 0), app(s, y)) -> 0 app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot 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(pred, app(s, x)) -> x app(app(minus, x), 0) -> x app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y)) app(app(quot, 0), app(s, y)) -> 0 app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot 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) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) 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(pred, app(s, x)) -> x app(app(minus, x), 0') -> x app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y)) app(app(quot, 0'), app(s, y)) -> 0' app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0') -> 0' encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0' encode_quot -> quot 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) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: app(pred, app(s, x)) -> x app(app(minus, x), 0') -> x app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y)) app(app(quot, 0'), app(s, y)) -> 0' app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0') -> 0' encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0' encode_quot -> quot encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app pred :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app s :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app minus :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app 0' :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app quot :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app map :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app nil :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app cons :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app filter :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app true :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app false :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encArg :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_pred :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_s :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_minus :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_0 :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_quot :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app hole_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app1_3 :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3 :: Nat -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app ---------------------------------------- (9) 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 ---------------------------------------- (10) Obligation: Innermost TRS: Rules: app(pred, app(s, x)) -> x app(app(minus, x), 0') -> x app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y)) app(app(quot, 0'), app(s, y)) -> 0' app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0') -> 0' encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0' encode_quot -> quot encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app pred :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app s :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app minus :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app 0' :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app quot :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app map :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app nil :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app cons :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app filter :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app true :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app false :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encArg :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_pred :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_s :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_minus :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_0 :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_quot :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app hole_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app1_3 :: pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3 :: Nat -> pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app Generator Equations: gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3(0) <=> pred gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3(+(x, 1)) <=> cons_app(pred, gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3(x)) The following defined symbols remain to be analysed: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3(n26572_3)) -> *3_3, rt in Omega(0) Induction Base: encArg(gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3(0)) Induction Step: encArg(gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3(+(n26572_3, 1))) ->_R^Omega(0) app(encArg(pred), encArg(gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3(n26572_3))) ->_R^Omega(0) app(pred, encArg(gen_pred:s:minus:0':quot:map:nil:cons:filter:filter2:true:false:cons_app2_3(n26572_3))) ->_IH app(pred, *3_3) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (12) BOUNDS(1, INF) ---------------------------------------- (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (14) 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(pred, app(s, x)) -> x app(app(minus, x), 0) -> x app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y)) app(app(quot, 0), app(s, y)) -> 0 app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (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(pred, app(s, x)) -> x app(app(minus, x), 0) -> x app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y)) app(app(quot, 0), app(s, y)) -> 0 app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot 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(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(pred, c_app(s, x)) -> x app(c_app(quot, 0), c_app(s, y)) -> 0 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(c_app(minus, x), c_app(s, y)) -> app(pred, app(app(minus, x), y)) app(c_app(minus, x), 0) -> x app(c_app(quot, c_app(s, x)), c_app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) The (relative) TRS S consists of the following rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot 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(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(pred, c_app(s, x)) -> x [1] app(c_app(quot, 0), c_app(s, y)) -> 0 [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(c_app(minus, x), c_app(s, y)) -> app(pred, app(app(minus, x), y)) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(quot, c_app(s, x)), c_app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) [1] encArg(pred) -> pred [0] encArg(s) -> s [0] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(quot) -> quot [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_pred -> pred [0] encode_s -> s [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_quot -> quot [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(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(pred, c_app(s, x)) -> x [1] app(c_app(quot, 0), c_app(s, y)) -> 0 [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(c_app(minus, x), c_app(s, y)) -> app(pred, app(app(minus, x), y)) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(quot, c_app(s, x)), c_app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) [1] encArg(pred) -> pred [0] encArg(s) -> s [0] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(quot) -> quot [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_pred -> pred [0] encode_s -> s [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_quot -> quot [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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app c_app :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app map :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app cons :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app 0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app filter2 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app true :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app filter :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app false :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app nil :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encArg :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app cons_app :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_app :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_map :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_nil :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_cons :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_filter :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_filter2 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_true :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app encode_false :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus: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_pred encode_s encode_minus encode_0 encode_quot 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_pred -> null_encode_pred [0] encode_s -> null_encode_s [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_quot -> null_encode_quot [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_pred, null_encode_s, null_encode_minus, null_encode_0, null_encode_quot, 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(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(pred, c_app(s, x)) -> x [1] app(c_app(quot, 0), c_app(s, y)) -> 0 [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(c_app(minus, x), c_app(s, y)) -> app(pred, app(app(minus, x), y)) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(quot, c_app(s, x)), c_app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) [1] encArg(pred) -> pred [0] encArg(s) -> s [0] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(quot) -> quot [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_pred -> pred [0] encode_s -> s [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_quot -> quot [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_pred -> null_encode_pred [0] encode_s -> null_encode_s [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_quot -> null_encode_quot [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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(pred, c_app(s, x)) -> x [1] app(c_app(quot, 0), c_app(s, y)) -> 0 [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(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, pred), c_app(c_app(cons, c_app(s, x9)), xs)) -> app(app(app(app(filter2, x9), pred), c_app(s, x9)), xs) [2] app(c_app(filter, c_app(quot, 0)), c_app(c_app(cons, c_app(s, y2)), xs)) -> app(app(app(app(filter2, 0), c_app(quot, 0)), c_app(s, y2)), 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, c_app(minus, x13)), c_app(c_app(cons, c_app(s, y3)), xs)) -> app(app(app(app(filter2, app(pred, app(app(minus, x13), y3))), c_app(minus, x13)), c_app(s, y3)), xs) [2] app(c_app(filter, c_app(minus, x14)), c_app(c_app(cons, 0), xs)) -> app(app(app(app(filter2, x14), c_app(minus, x14)), 0), xs) [2] app(c_app(filter, c_app(quot, c_app(s, x15))), c_app(c_app(cons, c_app(s, y4)), xs)) -> app(app(app(app(filter2, app(s, app(app(quot, app(app(minus, x15), y4)), app(s, y4)))), c_app(quot, c_app(s, x15))), c_app(s, y4)), 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(c_app(minus, x), c_app(s, y)) -> app(pred, app(c_app(minus, x), y)) [1] app(c_app(minus, x), c_app(s, y)) -> app(pred, app(null_app, y)) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(quot, c_app(s, x)), c_app(s, y)) -> app(s, app(app(quot, app(c_app(minus, x), y)), c_app(s, y))) [1] app(c_app(quot, c_app(s, x)), c_app(s, y)) -> app(s, app(app(quot, app(c_app(minus, x), y)), null_app)) [1] app(c_app(quot, c_app(s, x)), c_app(s, y)) -> app(s, app(app(quot, app(null_app, y)), c_app(s, y))) [1] app(c_app(quot, c_app(s, x)), c_app(s, y)) -> app(s, app(app(quot, app(null_app, y)), null_app)) [1] encArg(pred) -> pred [0] encArg(s) -> s [0] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(quot) -> quot [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_pred -> pred [0] encode_s -> s [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_quot -> quot [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_pred -> null_encode_pred [0] encode_s -> null_encode_s [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_quot -> null_encode_quot [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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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: map => 5 cons => 1 pred => 8 s => 10 quot => 9 0 => 0 filter2 => 4 true => 11 filter => 3 false => 2 nil => 7 minus => 6 null_encArg => 0 null_encode_app => 0 null_encode_pred => 0 null_encode_s => 0 null_encode_minus => 0 null_encode_0 => 0 null_encode_quot => 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 :|: x >= 0, z' = 1 + 10 + x, z = 8 app(z, z') -{ 1 }-> x :|: x >= 0, z = 1 + 6 + x, z' = 0 app(z, z') -{ 2 }-> app(app(app(app(4, x14), 1 + 6 + x14), 0), xs) :|: xs >= 0, z = 1 + 3 + (1 + 6 + x14), z' = 1 + (1 + 1 + 0) + xs, x14 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, x9), 8), 1 + 10 + x9), xs) :|: z' = 1 + (1 + 1 + (1 + 10 + x9)) + xs, xs >= 0, z = 1 + 3 + 8, 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) :|: z = 1 + 3 + (1 + 3 + f10), xs >= 0, xs2 >= 0, z' = 1 + (1 + 1 + (1 + (1 + 1 + x12) + xs2)) + xs, f10 >= 0, x12 >= 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, x >= 0, f7 >= 0, x11 >= 0, z = 1 + 3 + (1 + (1 + (1 + 4 + 2) + f7) + x11), 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 + 11) + f6) + x10), x), xs) :|: xs >= 0, x >= 0, f6 >= 0, z = 1 + 3 + (1 + (1 + (1 + 4 + 11) + f6) + x10), 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, x8 >= 0, z = 1 + 3 + (1 + 5 + f5), xs1 >= 0, f5 >= 0, z' = 1 + (1 + 1 + (1 + (1 + 1 + x8) + xs1)) + xs app(z, z') -{ 2 }-> app(app(app(app(4, app(8, app(app(6, x13), y3))), 1 + 6 + x13), 1 + 10 + y3), xs) :|: x13 >= 0, xs >= 0, z' = 1 + (1 + 1 + (1 + 10 + y3)) + xs, y3 >= 0, z = 1 + 3 + (1 + 6 + x13) app(z, z') -{ 2 }-> app(app(app(app(4, app(10, app(app(9, app(app(6, x15), y4)), app(10, y4)))), 1 + 9 + (1 + 10 + x15)), 1 + 10 + y4), xs) :|: z = 1 + 3 + (1 + 9 + (1 + 10 + x15)), xs >= 0, y4 >= 0, z' = 1 + (1 + 1 + (1 + 10 + y4)) + xs, x15 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, 7), 1 + 5 + f8), 7), xs) :|: z = 1 + 3 + (1 + 5 + f8), z' = 1 + (1 + 1 + 7) + xs, f8 >= 0, xs >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, 7), 1 + 3 + f9), 7), xs) :|: z' = 1 + (1 + 1 + 7) + xs, z = 1 + 3 + (1 + 3 + f9), xs >= 0, f9 >= 0 app(z, z') -{ 1 }-> app(app(app(app(4, 0), f), x), xs) :|: z = 1 + 3 + f, f >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 2 }-> app(app(app(app(4, 0), 1 + 9 + 0), 1 + 10 + y2), xs) :|: xs >= 0, z' = 1 + (1 + 1 + (1 + 10 + y2)) + xs, y2 >= 0, z = 1 + 3 + (1 + 9 + 0) app(z, z') -{ 1 }-> app(app(app(app(4, 1 + f + x), f), x), xs) :|: z = 1 + 3 + f, f >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs 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(8, app(0, y)) :|: x >= 0, y >= 0, z' = 1 + 10 + y, z = 1 + 6 + x app(z, z') -{ 1 }-> app(8, app(1 + 6 + x, y)) :|: x >= 0, y >= 0, z' = 1 + 10 + y, z = 1 + 6 + x app(z, z') -{ 1 }-> app(10, app(app(9, app(0, y)), 0)) :|: z = 1 + 9 + (1 + 10 + x), x >= 0, y >= 0, z' = 1 + 10 + y app(z, z') -{ 1 }-> app(10, app(app(9, app(0, y)), 1 + 10 + y)) :|: z = 1 + 9 + (1 + 10 + x), x >= 0, y >= 0, z' = 1 + 10 + y app(z, z') -{ 1 }-> app(10, app(app(9, app(1 + 6 + x, y)), 0)) :|: z = 1 + 9 + (1 + 10 + x), x >= 0, y >= 0, z' = 1 + 10 + y app(z, z') -{ 1 }-> app(10, app(app(9, app(1 + 6 + x, y)), 1 + 10 + y)) :|: z = 1 + 9 + (1 + 10 + x), x >= 0, y >= 0, z' = 1 + 10 + 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 + 11) + 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 + 11) + 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 + 11) + 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 + 11) + f) + x app(z, z') -{ 1 }-> 7 :|: f >= 0, z' = 7, z = 1 + 5 + f app(z, z') -{ 1 }-> 7 :|: z = 1 + 3 + f, f >= 0, z' = 7 app(z, z') -{ 1 }-> 0 :|: z = 1 + 9 + 0, y >= 0, z' = 1 + 10 + y app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 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 }-> 11 :|: z = 11 encArg(z) -{ 0 }-> 10 :|: z = 10 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_0 -{ 0 }-> 0 :|: encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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_minus -{ 0 }-> 6 :|: encode_minus -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 7 :|: encode_nil -{ 0 }-> 0 :|: encode_pred -{ 0 }-> 8 :|: encode_pred -{ 0 }-> 0 :|: encode_quot -{ 0 }-> 9 :|: encode_quot -{ 0 }-> 0 :|: encode_s -{ 0 }-> 10 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 11 :|: 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), 8), 1 + 10 + x9), xs) :|: z' = 1 + (1 + 1 + (1 + 10 + x9)) + xs, xs >= 0, z = 1 + 3 + 8, 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' = 1 + (1 + 1 + (1 + (1 + 1 + x12) + xs2)) + xs, z - 8 >= 0, x12 >= 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, x >= 0, f7 >= 0, x11 >= 0, z = 1 + 3 + (1 + (1 + (1 + 4 + 2) + f7) + x11), 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 + 11) + f6) + x10), x), xs) :|: xs >= 0, x >= 0, f6 >= 0, z = 1 + 3 + (1 + (1 + (1 + 4 + 11) + f6) + x10), 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, x8 >= 0, xs1 >= 0, z - 10 >= 0, z' = 1 + (1 + 1 + (1 + (1 + 1 + x8) + xs1)) + xs app(z, z') -{ 2 }-> app(app(app(app(4, app(8, app(app(6, z - 11), y3))), 1 + 6 + (z - 11)), 1 + 10 + y3), xs) :|: z - 11 >= 0, xs >= 0, z' = 1 + (1 + 1 + (1 + 10 + y3)) + xs, y3 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, app(10, app(app(9, app(app(6, z - 25), y4)), app(10, y4)))), 1 + 9 + (1 + 10 + (z - 25))), 1 + 10 + y4), xs) :|: xs >= 0, y4 >= 0, z' = 1 + (1 + 1 + (1 + 10 + y4)) + xs, z - 25 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, 7), 1 + 5 + (z - 10)), 7), z' - 10) :|: z - 10 >= 0, z' - 10 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, 7), 1 + 3 + (z - 8)), 7), z' - 10) :|: z' - 10 >= 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') -{ 2 }-> app(app(app(app(4, 0), 1 + 9 + 0), 1 + 10 + y2), xs) :|: xs >= 0, z' = 1 + (1 + 1 + (1 + 10 + y2)) + xs, y2 >= 0, z = 1 + 3 + (1 + 9 + 0) app(z, z') -{ 2 }-> app(app(app(app(4, z - 11), 1 + 6 + (z - 11)), 0), z' - 3) :|: z' - 3 >= 0, z - 11 >= 0 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(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(8, app(0, z' - 11)) :|: z - 7 >= 0, z' - 11 >= 0 app(z, z') -{ 1 }-> app(8, app(1 + 6 + (z - 7), z' - 11)) :|: z - 7 >= 0, z' - 11 >= 0 app(z, z') -{ 1 }-> app(10, app(app(9, app(0, z' - 11)), 0)) :|: z - 21 >= 0, z' - 11 >= 0 app(z, z') -{ 1 }-> app(10, app(app(9, app(0, z' - 11)), 1 + 10 + (z' - 11))) :|: z - 21 >= 0, z' - 11 >= 0 app(z, z') -{ 1 }-> app(10, app(app(9, app(1 + 6 + (z - 21), z' - 11)), 0)) :|: z - 21 >= 0, z' - 11 >= 0 app(z, z') -{ 1 }-> app(10, app(app(9, app(1 + 6 + (z - 21), z' - 11)), 1 + 10 + (z' - 11))) :|: z - 21 >= 0, z' - 11 >= 0 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 + 11) + f) + x app(z, z') -{ 1 }-> app(0, app(1 + 3 + f, z')) :|: f >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 4 + 11) + 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 + 11) + 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 + 11) + f) + x app(z, z') -{ 1 }-> 7 :|: z - 6 >= 0, z' = 7 app(z, z') -{ 1 }-> 7 :|: z - 4 >= 0, z' = 7 app(z, z') -{ 1 }-> 0 :|: z = 1 + 9 + 0, z' - 11 >= 0 app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 app(z, z') -{ 1 }-> z - 7 :|: z - 7 >= 0, z' = 0 app(z, z') -{ 1 }-> z' - 11 :|: z' - 11 >= 0, z = 8 app(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 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 }-> 11 :|: z = 11 encArg(z) -{ 0 }-> 10 :|: z = 10 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_0 -{ 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_minus -{ 0 }-> 6 :|: encode_minus -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 7 :|: encode_nil -{ 0 }-> 0 :|: encode_pred -{ 0 }-> 8 :|: encode_pred -{ 0 }-> 0 :|: encode_quot -{ 0 }-> 9 :|: encode_quot -{ 0 }-> 0 :|: encode_s -{ 0 }-> 10 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 11 :|: 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_pred -> null_encode_pred [0] encode_s -> null_encode_s [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_quot -> null_encode_quot [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_pred, null_encode_s, null_encode_minus, null_encode_0, null_encode_quot, 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(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1] app(pred, c_app(s, x)) -> x [1] app(c_app(quot, 0), c_app(s, y)) -> 0 [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(c_app(minus, x), c_app(s, y)) -> app(pred, app(app(minus, x), y)) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(quot, c_app(s, x)), c_app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y))) [1] encArg(pred) -> pred [0] encArg(s) -> s [0] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(quot) -> quot [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_pred -> pred [0] encode_s -> s [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_quot -> quot [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_pred -> null_encode_pred [0] encode_s -> null_encode_s [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_quot -> null_encode_quot [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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_pred :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_s :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_minus :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_0 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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_quot :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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 :: map:c_app:cons:pred:s:quot:0:filter2:true:filter:false:nil:minus:cons_app:null_encArg:null_encode_app:null_encode_pred:null_encode_s:null_encode_minus:null_encode_0:null_encode_quot: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: map => 5 cons => 1 pred => 8 s => 10 quot => 9 0 => 0 filter2 => 4 true => 11 filter => 3 false => 2 nil => 7 minus => 6 null_encArg => 0 null_encode_app => 0 null_encode_pred => 0 null_encode_s => 0 null_encode_minus => 0 null_encode_0 => 0 null_encode_quot => 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 :|: x >= 0, z' = 1 + 10 + x, z = 8 app(z, z') -{ 1 }-> x :|: x >= 0, z = 1 + 6 + x, z' = 0 app(z, z') -{ 1 }-> app(app(app(app(4, app(f, x)), f), x), xs) :|: z = 1 + 3 + f, f >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs 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 + 11) + 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(8, app(app(6, x), y)) :|: x >= 0, y >= 0, z' = 1 + 10 + y, z = 1 + 6 + x app(z, z') -{ 1 }-> app(10, app(app(9, app(app(6, x), y)), app(10, y))) :|: z = 1 + 9 + (1 + 10 + x), x >= 0, y >= 0, z' = 1 + 10 + y app(z, z') -{ 1 }-> 7 :|: f >= 0, z' = 7, z = 1 + 5 + f app(z, z') -{ 1 }-> 7 :|: z = 1 + 3 + f, f >= 0, z' = 7 app(z, z') -{ 1 }-> 0 :|: z = 1 + 9 + 0, y >= 0, z' = 1 + 10 + y app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 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 }-> 11 :|: z = 11 encArg(z) -{ 0 }-> 10 :|: z = 10 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_0 -{ 0 }-> 0 :|: encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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_minus -{ 0 }-> 6 :|: encode_minus -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 7 :|: encode_nil -{ 0 }-> 0 :|: encode_pred -{ 0 }-> 8 :|: encode_pred -{ 0 }-> 0 :|: encode_quot -{ 0 }-> 9 :|: encode_quot -{ 0 }-> 0 :|: encode_s -{ 0 }-> 10 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 11 :|: 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred) -> c ENCARG(s) -> c1 ENCARG(minus) -> c2 ENCARG(0) -> c3 ENCARG(quot) -> c4 ENCARG(map) -> c5 ENCARG(nil) -> c6 ENCARG(cons) -> c7 ENCARG(filter) -> c8 ENCARG(filter2) -> c9 ENCARG(true) -> c10 ENCARG(false) -> c11 ENCARG(cons_app(z0, z1)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c13(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_PRED -> c14 ENCODE_S -> c15 ENCODE_MINUS -> c16 ENCODE_0 -> c17 ENCODE_QUOT -> c18 ENCODE_MAP -> c19 ENCODE_NIL -> c20 ENCODE_CONS -> c21 ENCODE_FILTER -> c22 ENCODE_FILTER2 -> c23 ENCODE_TRUE -> c24 ENCODE_FALSE -> c25 APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1), APP(minus, z0)) APP(app(quot, 0), app(s, z0)) -> c29 APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1))), APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(quot, app(app(minus, z0), z1)), APP(app(minus, z0), z1), APP(minus, z0), APP(s, z1)) APP(app(map, z0), nil) -> c31 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(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) -> c33 APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(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) -> c35(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) -> c36(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1), APP(minus, z0)) APP(app(quot, 0), app(s, z0)) -> c29 APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1))), APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(quot, app(app(minus, z0), z1)), APP(app(minus, z0), z1), APP(minus, z0), APP(s, z1)) APP(app(map, z0), nil) -> c31 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(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) -> c33 APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(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) -> c35(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) -> c36(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_pred, encode_s, encode_minus, encode_0, encode_quot, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, ENCODE_PRED, ENCODE_S, ENCODE_MINUS, ENCODE_0, ENCODE_QUOT, 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, c11, c12_3, c13_3, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28_3, c29, c30_6, c31, c32_5, c33, c34_5, c35_4, c36_2 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 24 trailing nodes: ENCODE_QUOT -> c18 ENCARG(nil) -> c6 ENCODE_0 -> c17 ENCODE_FILTER2 -> c23 ENCODE_FILTER -> c22 ENCODE_PRED -> c14 ENCARG(true) -> c10 ENCARG(0) -> c3 ENCARG(s) -> c1 ENCARG(filter2) -> c9 ENCODE_MAP -> c19 ENCARG(quot) -> c4 ENCARG(pred) -> c ENCODE_FALSE -> c25 ENCODE_MINUS -> c16 ENCARG(false) -> c11 ENCODE_NIL -> c20 ENCARG(cons) -> c7 ENCODE_TRUE -> c24 ENCARG(filter) -> c8 ENCODE_CONS -> c21 ENCARG(map) -> c5 ENCARG(minus) -> c2 ENCODE_S -> c15 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c13(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1), APP(minus, z0)) APP(app(quot, 0), app(s, z0)) -> c29 APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1))), APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(quot, app(app(minus, z0), z1)), APP(app(minus, z0), z1), APP(minus, z0), APP(s, z1)) APP(app(map, z0), nil) -> c31 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(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) -> c33 APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(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) -> c35(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) -> c36(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1), APP(minus, z0)) APP(app(quot, 0), app(s, z0)) -> c29 APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1))), APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(quot, app(app(minus, z0), z1)), APP(app(minus, z0), z1), APP(minus, z0), APP(s, z1)) APP(app(map, z0), nil) -> c31 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(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) -> c33 APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(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) -> c35(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) -> c36(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_pred, encode_s, encode_minus, encode_0, encode_quot, 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: c12_3, c13_3, c26, c27, c28_3, c29, c30_6, c31, c32_5, c33, c34_5, c35_4, c36_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 15 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c13(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_pred, encode_s, encode_minus, encode_0, encode_quot, 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: c12_3, c13_3, c26, c27, c29, c31, c33, c28_2, c30_2, c32_2, c34_2, c35_1, c36_1 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_pred, encode_s, encode_minus, encode_0, encode_quot, 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: c12_3, c26, c27, c29, c31, c33, c28_2, c30_2, c32_2, c34_2, c35_1, c36_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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_pred, encode_s, encode_minus, encode_0, encode_quot, 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: c12_3, c26, c27, c29, c31, c33, c28_2, c30_2, c32_2, c34_2, c35_1, c36_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_pred -> pred encode_s -> s encode_minus -> minus encode_0 -> 0 encode_quot -> quot 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(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(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: c12_3, c26, c27, c29, c31, c33, c28_2, c30_2, c32_2, c34_2, c35_1, c36_1, c_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0), ENCARG(pred)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0), ENCARG(s)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0), ENCARG(minus)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0), ENCARG(quot)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(pred), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(APP(s, encArg(x1)), ENCARG(s), ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(APP(minus, encArg(x1)), ENCARG(minus), ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(APP(quot, encArg(x1)), ENCARG(quot), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0), ENCARG(pred)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0), ENCARG(s)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0), ENCARG(minus)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0), ENCARG(quot)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(pred), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(APP(s, encArg(x1)), ENCARG(s), ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(APP(minus, encArg(x1)), ENCARG(minus), ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(APP(quot, encArg(x1)), ENCARG(quot), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(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: c26, c27, c29, c31, c33, c28_2, c30_2, c32_2, c34_2, c35_1, c36_1, c_1, c12_3 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 35 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(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: c26, c27, c29, c31, c33, c28_2, c30_2, c32_2, c34_2, c35_1, c36_1, c_1, c12_3, c12_2, c12_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(minus, z0), app(s, z1)) -> c28(APP(pred, app(app(minus, z0), z1)), APP(app(minus, z0), z1)) by APP(app(minus, z0), app(s, 0)) -> c28(APP(pred, z0), APP(app(minus, z0), 0)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c28(APP(pred, z0), APP(app(minus, z0), 0)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c28(APP(pred, z0), APP(app(minus, z0), 0)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c26, c27, c29, c31, c33, c30_2, c32_2, c34_2, c35_1, c36_1, c_1, c12_3, c12_2, c12_1, c28_2 ---------------------------------------- (53) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c26, c27, c29, c31, c33, c30_2, c32_2, c34_2, c35_1, c36_1, c_1, c12_3, c12_2, c12_1, c28_2, c1_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(quot, app(s, z0)), app(s, z1)) -> c30(APP(app(quot, app(app(minus, z0), z1)), app(s, z1)), APP(app(minus, z0), z1)) by APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c26, c27, c29, c31, c33, c32_2, c34_2, c35_1, c36_1, c_1, c12_3, c12_2, c12_1, c28_2, c1_1, c30_2 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, z0), app(app(cons, z1), z2)) -> c34(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) by APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, 0), app(quot, 0)), app(s, z0)), x2), APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1)))), app(quot, app(s, z0))), app(s, z1)), x2), APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, 0), app(quot, 0)), app(s, z0)), x2), APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1)))), app(quot, app(s, z0))), app(s, z1)), x2), APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, 0), app(quot, 0)), app(s, z0)), x2), APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1)))), app(quot, app(s, z0))), app(s, z1)), x2), APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c_1, c12_3, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c_1, c12_3, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) by ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(minus, x1) -> c(APP(minus, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(quot, x1) -> c(APP(quot, 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))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(minus, x1) -> c(APP(minus, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(quot, x1) -> c(APP(quot, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_3, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(quot, x1) -> c(APP(quot, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(minus, x1) -> c(APP(minus, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_3, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by ENCARG(cons_app(x0, cons_app(x1, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(pred), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(APP(s, app(encArg(x1), encArg(x2))), ENCARG(s), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(APP(minus, app(encArg(x1), encArg(x2))), ENCARG(minus), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(APP(quot, app(encArg(x1), encArg(x2))), ENCARG(quot), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(pred), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(APP(s, app(encArg(x1), encArg(x2))), ENCARG(s), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(APP(minus, app(encArg(x1), encArg(x2))), ENCARG(minus), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(APP(quot, app(encArg(x1), encArg(x2))), ENCARG(quot), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_3, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 23 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_3, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by ENCARG(cons_app(cons_app(x0, x1), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1)), ENCARG(pred)) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1)), ENCARG(s)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1)), ENCARG(minus)) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0)) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1)), ENCARG(quot)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(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))) -> c12(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, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(APP(app(s, encArg(x1)), encArg(x2)), ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(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)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, x1), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1)), ENCARG(pred)) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1)), ENCARG(s)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1)), ENCARG(minus)) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0)) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1)), ENCARG(quot)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(APP(app(s, encArg(x1)), encArg(x2)), ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(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)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 19 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, pred)) -> c12(APP(encArg(x0), pred), ENCARG(x0)) by ENCARG(cons_app(pred, pred)) -> c12(APP(pred, pred), ENCARG(pred)) ENCARG(cons_app(s, pred)) -> c12(APP(s, pred), ENCARG(s)) ENCARG(cons_app(minus, pred)) -> c12(APP(minus, pred), ENCARG(minus)) ENCARG(cons_app(0, pred)) -> c12(APP(0, pred), ENCARG(0)) ENCARG(cons_app(quot, pred)) -> c12(APP(quot, pred), ENCARG(quot)) ENCARG(cons_app(map, pred)) -> c12(APP(map, pred), ENCARG(map)) ENCARG(cons_app(nil, pred)) -> c12(APP(nil, pred), ENCARG(nil)) ENCARG(cons_app(cons, pred)) -> c12(APP(cons, pred), ENCARG(cons)) ENCARG(cons_app(filter, pred)) -> c12(APP(filter, pred), ENCARG(filter)) ENCARG(cons_app(filter2, pred)) -> c12(APP(filter2, pred), ENCARG(filter2)) ENCARG(cons_app(true, pred)) -> c12(APP(true, pred), ENCARG(true)) ENCARG(cons_app(false, pred)) -> c12(APP(false, pred), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), pred)) -> c12(APP(app(encArg(z0), encArg(z1)), pred), ENCARG(cons_app(z0, z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, pred)) -> c12(APP(pred, pred), ENCARG(pred)) ENCARG(cons_app(s, pred)) -> c12(APP(s, pred), ENCARG(s)) ENCARG(cons_app(minus, pred)) -> c12(APP(minus, pred), ENCARG(minus)) ENCARG(cons_app(0, pred)) -> c12(APP(0, pred), ENCARG(0)) ENCARG(cons_app(quot, pred)) -> c12(APP(quot, pred), ENCARG(quot)) ENCARG(cons_app(map, pred)) -> c12(APP(map, pred), ENCARG(map)) ENCARG(cons_app(nil, pred)) -> c12(APP(nil, pred), ENCARG(nil)) ENCARG(cons_app(cons, pred)) -> c12(APP(cons, pred), ENCARG(cons)) ENCARG(cons_app(filter, pred)) -> c12(APP(filter, pred), ENCARG(filter)) ENCARG(cons_app(filter2, pred)) -> c12(APP(filter2, pred), ENCARG(filter2)) ENCARG(cons_app(true, pred)) -> c12(APP(true, pred), ENCARG(true)) ENCARG(cons_app(false, pred)) -> c12(APP(false, pred), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(filter, pred)) -> c12(APP(filter, pred), ENCARG(filter)) ENCARG(cons_app(cons, pred)) -> c12(APP(cons, pred), ENCARG(cons)) ENCARG(cons_app(0, pred)) -> c12(APP(0, pred), ENCARG(0)) ENCARG(cons_app(filter2, pred)) -> c12(APP(filter2, pred), ENCARG(filter2)) ENCARG(cons_app(false, pred)) -> c12(APP(false, pred), ENCARG(false)) ENCARG(cons_app(quot, pred)) -> c12(APP(quot, pred), ENCARG(quot)) ENCARG(cons_app(s, pred)) -> c12(APP(s, pred), ENCARG(s)) ENCARG(cons_app(pred, pred)) -> c12(APP(pred, pred), ENCARG(pred)) ENCARG(cons_app(true, pred)) -> c12(APP(true, pred), ENCARG(true)) ENCARG(cons_app(nil, pred)) -> c12(APP(nil, pred), ENCARG(nil)) ENCARG(cons_app(minus, pred)) -> c12(APP(minus, pred), ENCARG(minus)) ENCARG(cons_app(map, pred)) -> c12(APP(map, pred), ENCARG(map)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) by ENCARG(cons_app(pred, s)) -> c12(APP(pred, s), ENCARG(pred)) ENCARG(cons_app(s, s)) -> c12(APP(s, s), ENCARG(s)) ENCARG(cons_app(minus, s)) -> c12(APP(minus, s), ENCARG(minus)) ENCARG(cons_app(0, s)) -> c12(APP(0, s), ENCARG(0)) ENCARG(cons_app(quot, s)) -> c12(APP(quot, s), ENCARG(quot)) ENCARG(cons_app(map, s)) -> c12(APP(map, s), ENCARG(map)) ENCARG(cons_app(nil, s)) -> c12(APP(nil, s), ENCARG(nil)) ENCARG(cons_app(cons, s)) -> c12(APP(cons, s), ENCARG(cons)) ENCARG(cons_app(filter, s)) -> c12(APP(filter, s), ENCARG(filter)) ENCARG(cons_app(filter2, s)) -> c12(APP(filter2, s), ENCARG(filter2)) ENCARG(cons_app(true, s)) -> c12(APP(true, s), ENCARG(true)) ENCARG(cons_app(false, s)) -> c12(APP(false, s), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), s)) -> c12(APP(app(encArg(z0), encArg(z1)), s), ENCARG(cons_app(z0, z1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, s)) -> c12(APP(pred, s), ENCARG(pred)) ENCARG(cons_app(s, s)) -> c12(APP(s, s), ENCARG(s)) ENCARG(cons_app(minus, s)) -> c12(APP(minus, s), ENCARG(minus)) ENCARG(cons_app(0, s)) -> c12(APP(0, s), ENCARG(0)) ENCARG(cons_app(quot, s)) -> c12(APP(quot, s), ENCARG(quot)) ENCARG(cons_app(map, s)) -> c12(APP(map, s), ENCARG(map)) ENCARG(cons_app(nil, s)) -> c12(APP(nil, s), ENCARG(nil)) ENCARG(cons_app(cons, s)) -> c12(APP(cons, s), ENCARG(cons)) ENCARG(cons_app(filter, s)) -> c12(APP(filter, s), ENCARG(filter)) ENCARG(cons_app(filter2, s)) -> c12(APP(filter2, s), ENCARG(filter2)) ENCARG(cons_app(true, s)) -> c12(APP(true, s), ENCARG(true)) ENCARG(cons_app(false, s)) -> c12(APP(false, s), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(filter2, s)) -> c12(APP(filter2, s), ENCARG(filter2)) ENCARG(cons_app(s, s)) -> c12(APP(s, s), ENCARG(s)) ENCARG(cons_app(cons, s)) -> c12(APP(cons, s), ENCARG(cons)) ENCARG(cons_app(quot, s)) -> c12(APP(quot, s), ENCARG(quot)) ENCARG(cons_app(0, s)) -> c12(APP(0, s), ENCARG(0)) ENCARG(cons_app(minus, s)) -> c12(APP(minus, s), ENCARG(minus)) ENCARG(cons_app(false, s)) -> c12(APP(false, s), ENCARG(false)) ENCARG(cons_app(filter, s)) -> c12(APP(filter, s), ENCARG(filter)) ENCARG(cons_app(true, s)) -> c12(APP(true, s), ENCARG(true)) ENCARG(cons_app(nil, s)) -> c12(APP(nil, s), ENCARG(nil)) ENCARG(cons_app(pred, s)) -> c12(APP(pred, s), ENCARG(pred)) ENCARG(cons_app(map, s)) -> c12(APP(map, s), ENCARG(map)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) by ENCARG(cons_app(pred, minus)) -> c12(APP(pred, minus), ENCARG(pred)) ENCARG(cons_app(s, minus)) -> c12(APP(s, minus), ENCARG(s)) ENCARG(cons_app(minus, minus)) -> c12(APP(minus, minus), ENCARG(minus)) ENCARG(cons_app(0, minus)) -> c12(APP(0, minus), ENCARG(0)) ENCARG(cons_app(quot, minus)) -> c12(APP(quot, minus), ENCARG(quot)) ENCARG(cons_app(map, minus)) -> c12(APP(map, minus), ENCARG(map)) ENCARG(cons_app(nil, minus)) -> c12(APP(nil, minus), ENCARG(nil)) ENCARG(cons_app(cons, minus)) -> c12(APP(cons, minus), ENCARG(cons)) ENCARG(cons_app(filter, minus)) -> c12(APP(filter, minus), ENCARG(filter)) ENCARG(cons_app(filter2, minus)) -> c12(APP(filter2, minus), ENCARG(filter2)) ENCARG(cons_app(true, minus)) -> c12(APP(true, minus), ENCARG(true)) ENCARG(cons_app(false, minus)) -> c12(APP(false, minus), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), minus)) -> c12(APP(app(encArg(z0), encArg(z1)), minus), ENCARG(cons_app(z0, z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, minus)) -> c12(APP(pred, minus), ENCARG(pred)) ENCARG(cons_app(s, minus)) -> c12(APP(s, minus), ENCARG(s)) ENCARG(cons_app(minus, minus)) -> c12(APP(minus, minus), ENCARG(minus)) ENCARG(cons_app(0, minus)) -> c12(APP(0, minus), ENCARG(0)) ENCARG(cons_app(quot, minus)) -> c12(APP(quot, minus), ENCARG(quot)) ENCARG(cons_app(map, minus)) -> c12(APP(map, minus), ENCARG(map)) ENCARG(cons_app(nil, minus)) -> c12(APP(nil, minus), ENCARG(nil)) ENCARG(cons_app(cons, minus)) -> c12(APP(cons, minus), ENCARG(cons)) ENCARG(cons_app(filter, minus)) -> c12(APP(filter, minus), ENCARG(filter)) ENCARG(cons_app(filter2, minus)) -> c12(APP(filter2, minus), ENCARG(filter2)) ENCARG(cons_app(true, minus)) -> c12(APP(true, minus), ENCARG(true)) ENCARG(cons_app(false, minus)) -> c12(APP(false, minus), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(false, minus)) -> c12(APP(false, minus), ENCARG(false)) ENCARG(cons_app(true, minus)) -> c12(APP(true, minus), ENCARG(true)) ENCARG(cons_app(filter2, minus)) -> c12(APP(filter2, minus), ENCARG(filter2)) ENCARG(cons_app(pred, minus)) -> c12(APP(pred, minus), ENCARG(pred)) ENCARG(cons_app(cons, minus)) -> c12(APP(cons, minus), ENCARG(cons)) ENCARG(cons_app(0, minus)) -> c12(APP(0, minus), ENCARG(0)) ENCARG(cons_app(filter, minus)) -> c12(APP(filter, minus), ENCARG(filter)) ENCARG(cons_app(quot, minus)) -> c12(APP(quot, minus), ENCARG(quot)) ENCARG(cons_app(nil, minus)) -> c12(APP(nil, minus), ENCARG(nil)) ENCARG(cons_app(s, minus)) -> c12(APP(s, minus), ENCARG(s)) ENCARG(cons_app(map, minus)) -> c12(APP(map, minus), ENCARG(map)) ENCARG(cons_app(minus, minus)) -> c12(APP(minus, minus), ENCARG(minus)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) by ENCARG(cons_app(pred, 0)) -> c12(APP(pred, 0), ENCARG(pred)) ENCARG(cons_app(s, 0)) -> c12(APP(s, 0), ENCARG(s)) ENCARG(cons_app(minus, 0)) -> c12(APP(minus, 0), ENCARG(minus)) ENCARG(cons_app(0, 0)) -> c12(APP(0, 0), ENCARG(0)) ENCARG(cons_app(quot, 0)) -> c12(APP(quot, 0), ENCARG(quot)) ENCARG(cons_app(map, 0)) -> c12(APP(map, 0), ENCARG(map)) ENCARG(cons_app(nil, 0)) -> c12(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(cons, 0)) -> c12(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(filter, 0)) -> c12(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(filter2, 0)) -> c12(APP(filter2, 0), ENCARG(filter2)) ENCARG(cons_app(true, 0)) -> c12(APP(true, 0), ENCARG(true)) ENCARG(cons_app(false, 0)) -> c12(APP(false, 0), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), 0)) -> c12(APP(app(encArg(z0), encArg(z1)), 0), ENCARG(cons_app(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, 0)) -> c12(APP(pred, 0), ENCARG(pred)) ENCARG(cons_app(s, 0)) -> c12(APP(s, 0), ENCARG(s)) ENCARG(cons_app(minus, 0)) -> c12(APP(minus, 0), ENCARG(minus)) ENCARG(cons_app(0, 0)) -> c12(APP(0, 0), ENCARG(0)) ENCARG(cons_app(quot, 0)) -> c12(APP(quot, 0), ENCARG(quot)) ENCARG(cons_app(map, 0)) -> c12(APP(map, 0), ENCARG(map)) ENCARG(cons_app(nil, 0)) -> c12(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(cons, 0)) -> c12(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(filter, 0)) -> c12(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(filter2, 0)) -> c12(APP(filter2, 0), ENCARG(filter2)) ENCARG(cons_app(true, 0)) -> c12(APP(true, 0), ENCARG(true)) ENCARG(cons_app(false, 0)) -> c12(APP(false, 0), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(filter, 0)) -> c12(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(cons, 0)) -> c12(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(0, 0)) -> c12(APP(0, 0), ENCARG(0)) ENCARG(cons_app(nil, 0)) -> c12(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(filter2, 0)) -> c12(APP(filter2, 0), ENCARG(filter2)) ENCARG(cons_app(map, 0)) -> c12(APP(map, 0), ENCARG(map)) ENCARG(cons_app(true, 0)) -> c12(APP(true, 0), ENCARG(true)) ENCARG(cons_app(minus, 0)) -> c12(APP(minus, 0), ENCARG(minus)) ENCARG(cons_app(false, 0)) -> c12(APP(false, 0), ENCARG(false)) ENCARG(cons_app(s, 0)) -> c12(APP(s, 0), ENCARG(s)) ENCARG(cons_app(quot, 0)) -> c12(APP(quot, 0), ENCARG(quot)) ENCARG(cons_app(pred, 0)) -> c12(APP(pred, 0), ENCARG(pred)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, quot)) -> c12(APP(encArg(x0), quot), ENCARG(x0)) by ENCARG(cons_app(pred, quot)) -> c12(APP(pred, quot), ENCARG(pred)) ENCARG(cons_app(s, quot)) -> c12(APP(s, quot), ENCARG(s)) ENCARG(cons_app(minus, quot)) -> c12(APP(minus, quot), ENCARG(minus)) ENCARG(cons_app(0, quot)) -> c12(APP(0, quot), ENCARG(0)) ENCARG(cons_app(quot, quot)) -> c12(APP(quot, quot), ENCARG(quot)) ENCARG(cons_app(map, quot)) -> c12(APP(map, quot), ENCARG(map)) ENCARG(cons_app(nil, quot)) -> c12(APP(nil, quot), ENCARG(nil)) ENCARG(cons_app(cons, quot)) -> c12(APP(cons, quot), ENCARG(cons)) ENCARG(cons_app(filter, quot)) -> c12(APP(filter, quot), ENCARG(filter)) ENCARG(cons_app(filter2, quot)) -> c12(APP(filter2, quot), ENCARG(filter2)) ENCARG(cons_app(true, quot)) -> c12(APP(true, quot), ENCARG(true)) ENCARG(cons_app(false, quot)) -> c12(APP(false, quot), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), quot)) -> c12(APP(app(encArg(z0), encArg(z1)), quot), ENCARG(cons_app(z0, z1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, quot)) -> c12(APP(pred, quot), ENCARG(pred)) ENCARG(cons_app(s, quot)) -> c12(APP(s, quot), ENCARG(s)) ENCARG(cons_app(minus, quot)) -> c12(APP(minus, quot), ENCARG(minus)) ENCARG(cons_app(0, quot)) -> c12(APP(0, quot), ENCARG(0)) ENCARG(cons_app(quot, quot)) -> c12(APP(quot, quot), ENCARG(quot)) ENCARG(cons_app(map, quot)) -> c12(APP(map, quot), ENCARG(map)) ENCARG(cons_app(nil, quot)) -> c12(APP(nil, quot), ENCARG(nil)) ENCARG(cons_app(cons, quot)) -> c12(APP(cons, quot), ENCARG(cons)) ENCARG(cons_app(filter, quot)) -> c12(APP(filter, quot), ENCARG(filter)) ENCARG(cons_app(filter2, quot)) -> c12(APP(filter2, quot), ENCARG(filter2)) ENCARG(cons_app(true, quot)) -> c12(APP(true, quot), ENCARG(true)) ENCARG(cons_app(false, quot)) -> c12(APP(false, quot), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(false, quot)) -> c12(APP(false, quot), ENCARG(false)) ENCARG(cons_app(filter, quot)) -> c12(APP(filter, quot), ENCARG(filter)) ENCARG(cons_app(map, quot)) -> c12(APP(map, quot), ENCARG(map)) ENCARG(cons_app(filter2, quot)) -> c12(APP(filter2, quot), ENCARG(filter2)) ENCARG(cons_app(nil, quot)) -> c12(APP(nil, quot), ENCARG(nil)) ENCARG(cons_app(pred, quot)) -> c12(APP(pred, quot), ENCARG(pred)) ENCARG(cons_app(true, quot)) -> c12(APP(true, quot), ENCARG(true)) ENCARG(cons_app(s, quot)) -> c12(APP(s, quot), ENCARG(s)) ENCARG(cons_app(quot, quot)) -> c12(APP(quot, quot), ENCARG(quot)) ENCARG(cons_app(cons, quot)) -> c12(APP(cons, quot), ENCARG(cons)) ENCARG(cons_app(0, quot)) -> c12(APP(0, quot), ENCARG(0)) ENCARG(cons_app(minus, quot)) -> c12(APP(minus, quot), ENCARG(minus)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) by ENCARG(cons_app(pred, map)) -> c12(APP(pred, map), ENCARG(pred)) ENCARG(cons_app(s, map)) -> c12(APP(s, map), ENCARG(s)) ENCARG(cons_app(minus, map)) -> c12(APP(minus, map), ENCARG(minus)) ENCARG(cons_app(0, map)) -> c12(APP(0, map), ENCARG(0)) ENCARG(cons_app(quot, map)) -> c12(APP(quot, map), ENCARG(quot)) ENCARG(cons_app(map, map)) -> c12(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c12(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c12(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c12(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c12(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c12(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c12(APP(false, map), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), map)) -> c12(APP(app(encArg(z0), encArg(z1)), map), ENCARG(cons_app(z0, z1))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, map)) -> c12(APP(pred, map), ENCARG(pred)) ENCARG(cons_app(s, map)) -> c12(APP(s, map), ENCARG(s)) ENCARG(cons_app(minus, map)) -> c12(APP(minus, map), ENCARG(minus)) ENCARG(cons_app(0, map)) -> c12(APP(0, map), ENCARG(0)) ENCARG(cons_app(quot, map)) -> c12(APP(quot, map), ENCARG(quot)) ENCARG(cons_app(map, map)) -> c12(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c12(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c12(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c12(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c12(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c12(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c12(APP(false, map), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(nil, map)) -> c12(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(s, map)) -> c12(APP(s, map), ENCARG(s)) ENCARG(cons_app(filter2, map)) -> c12(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(cons, map)) -> c12(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(0, map)) -> c12(APP(0, map), ENCARG(0)) ENCARG(cons_app(minus, map)) -> c12(APP(minus, map), ENCARG(minus)) ENCARG(cons_app(filter, map)) -> c12(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(map, map)) -> c12(APP(map, map), ENCARG(map)) ENCARG(cons_app(false, map)) -> c12(APP(false, map), ENCARG(false)) ENCARG(cons_app(pred, map)) -> c12(APP(pred, map), ENCARG(pred)) ENCARG(cons_app(true, map)) -> c12(APP(true, map), ENCARG(true)) ENCARG(cons_app(quot, map)) -> c12(APP(quot, map), ENCARG(quot)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) by ENCARG(cons_app(pred, nil)) -> c12(APP(pred, nil), ENCARG(pred)) ENCARG(cons_app(s, nil)) -> c12(APP(s, nil), ENCARG(s)) ENCARG(cons_app(minus, nil)) -> c12(APP(minus, nil), ENCARG(minus)) ENCARG(cons_app(0, nil)) -> c12(APP(0, nil), ENCARG(0)) ENCARG(cons_app(quot, nil)) -> c12(APP(quot, nil), ENCARG(quot)) ENCARG(cons_app(map, nil)) -> c12(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c12(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c12(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c12(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c12(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c12(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c12(APP(false, nil), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), nil)) -> c12(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, nil)) -> c12(APP(pred, nil), ENCARG(pred)) ENCARG(cons_app(s, nil)) -> c12(APP(s, nil), ENCARG(s)) ENCARG(cons_app(minus, nil)) -> c12(APP(minus, nil), ENCARG(minus)) ENCARG(cons_app(0, nil)) -> c12(APP(0, nil), ENCARG(0)) ENCARG(cons_app(quot, nil)) -> c12(APP(quot, nil), ENCARG(quot)) ENCARG(cons_app(map, nil)) -> c12(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c12(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c12(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c12(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c12(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c12(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c12(APP(false, nil), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(map, nil)) -> c12(APP(map, nil), ENCARG(map)) ENCARG(cons_app(quot, nil)) -> c12(APP(quot, nil), ENCARG(quot)) ENCARG(cons_app(true, nil)) -> c12(APP(true, nil), ENCARG(true)) ENCARG(cons_app(filter2, nil)) -> c12(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(pred, nil)) -> c12(APP(pred, nil), ENCARG(pred)) ENCARG(cons_app(filter, nil)) -> c12(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(nil, nil)) -> c12(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(s, nil)) -> c12(APP(s, nil), ENCARG(s)) ENCARG(cons_app(false, nil)) -> c12(APP(false, nil), ENCARG(false)) ENCARG(cons_app(minus, nil)) -> c12(APP(minus, nil), ENCARG(minus)) ENCARG(cons_app(cons, nil)) -> c12(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(0, nil)) -> c12(APP(0, nil), ENCARG(0)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) by ENCARG(cons_app(pred, cons)) -> c12(APP(pred, cons), ENCARG(pred)) ENCARG(cons_app(s, cons)) -> c12(APP(s, cons), ENCARG(s)) ENCARG(cons_app(minus, cons)) -> c12(APP(minus, cons), ENCARG(minus)) ENCARG(cons_app(0, cons)) -> c12(APP(0, cons), ENCARG(0)) ENCARG(cons_app(quot, cons)) -> c12(APP(quot, cons), ENCARG(quot)) ENCARG(cons_app(map, cons)) -> c12(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c12(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c12(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c12(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c12(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c12(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c12(APP(false, cons), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), cons)) -> c12(APP(app(encArg(z0), encArg(z1)), cons), ENCARG(cons_app(z0, z1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, cons)) -> c12(APP(pred, cons), ENCARG(pred)) ENCARG(cons_app(s, cons)) -> c12(APP(s, cons), ENCARG(s)) ENCARG(cons_app(minus, cons)) -> c12(APP(minus, cons), ENCARG(minus)) ENCARG(cons_app(0, cons)) -> c12(APP(0, cons), ENCARG(0)) ENCARG(cons_app(quot, cons)) -> c12(APP(quot, cons), ENCARG(quot)) ENCARG(cons_app(map, cons)) -> c12(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c12(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c12(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c12(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c12(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c12(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c12(APP(false, cons), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(false, cons)) -> c12(APP(false, cons), ENCARG(false)) ENCARG(cons_app(true, cons)) -> c12(APP(true, cons), ENCARG(true)) ENCARG(cons_app(filter, cons)) -> c12(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(pred, cons)) -> c12(APP(pred, cons), ENCARG(pred)) ENCARG(cons_app(filter2, cons)) -> c12(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(nil, cons)) -> c12(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(s, cons)) -> c12(APP(s, cons), ENCARG(s)) ENCARG(cons_app(cons, cons)) -> c12(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(0, cons)) -> c12(APP(0, cons), ENCARG(0)) ENCARG(cons_app(quot, cons)) -> c12(APP(quot, cons), ENCARG(quot)) ENCARG(cons_app(minus, cons)) -> c12(APP(minus, cons), ENCARG(minus)) ENCARG(cons_app(map, cons)) -> c12(APP(map, cons), ENCARG(map)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) by ENCARG(cons_app(pred, filter)) -> c12(APP(pred, filter), ENCARG(pred)) ENCARG(cons_app(s, filter)) -> c12(APP(s, filter), ENCARG(s)) ENCARG(cons_app(minus, filter)) -> c12(APP(minus, filter), ENCARG(minus)) ENCARG(cons_app(0, filter)) -> c12(APP(0, filter), ENCARG(0)) ENCARG(cons_app(quot, filter)) -> c12(APP(quot, filter), ENCARG(quot)) ENCARG(cons_app(map, filter)) -> c12(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c12(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c12(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c12(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c12(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c12(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c12(APP(false, filter), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter)) -> c12(APP(app(encArg(z0), encArg(z1)), filter), ENCARG(cons_app(z0, z1))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, filter)) -> c12(APP(pred, filter), ENCARG(pred)) ENCARG(cons_app(s, filter)) -> c12(APP(s, filter), ENCARG(s)) ENCARG(cons_app(minus, filter)) -> c12(APP(minus, filter), ENCARG(minus)) ENCARG(cons_app(0, filter)) -> c12(APP(0, filter), ENCARG(0)) ENCARG(cons_app(quot, filter)) -> c12(APP(quot, filter), ENCARG(quot)) ENCARG(cons_app(map, filter)) -> c12(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c12(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c12(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c12(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c12(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c12(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c12(APP(false, filter), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(s, filter)) -> c12(APP(s, filter), ENCARG(s)) ENCARG(cons_app(map, filter)) -> c12(APP(map, filter), ENCARG(map)) ENCARG(cons_app(true, filter)) -> c12(APP(true, filter), ENCARG(true)) ENCARG(cons_app(cons, filter)) -> c12(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter2, filter)) -> c12(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(0, filter)) -> c12(APP(0, filter), ENCARG(0)) ENCARG(cons_app(false, filter)) -> c12(APP(false, filter), ENCARG(false)) ENCARG(cons_app(quot, filter)) -> c12(APP(quot, filter), ENCARG(quot)) ENCARG(cons_app(minus, filter)) -> c12(APP(minus, filter), ENCARG(minus)) ENCARG(cons_app(pred, filter)) -> c12(APP(pred, filter), ENCARG(pred)) ENCARG(cons_app(filter, filter)) -> c12(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(nil, filter)) -> c12(APP(nil, filter), ENCARG(nil)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) by ENCARG(cons_app(pred, filter2)) -> c12(APP(pred, filter2), ENCARG(pred)) ENCARG(cons_app(s, filter2)) -> c12(APP(s, filter2), ENCARG(s)) ENCARG(cons_app(minus, filter2)) -> c12(APP(minus, filter2), ENCARG(minus)) ENCARG(cons_app(0, filter2)) -> c12(APP(0, filter2), ENCARG(0)) ENCARG(cons_app(quot, filter2)) -> c12(APP(quot, filter2), ENCARG(quot)) ENCARG(cons_app(map, filter2)) -> c12(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c12(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c12(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c12(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c12(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c12(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c12(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter2)) -> c12(APP(app(encArg(z0), encArg(z1)), filter2), ENCARG(cons_app(z0, z1))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, filter2)) -> c12(APP(pred, filter2), ENCARG(pred)) ENCARG(cons_app(s, filter2)) -> c12(APP(s, filter2), ENCARG(s)) ENCARG(cons_app(minus, filter2)) -> c12(APP(minus, filter2), ENCARG(minus)) ENCARG(cons_app(0, filter2)) -> c12(APP(0, filter2), ENCARG(0)) ENCARG(cons_app(quot, filter2)) -> c12(APP(quot, filter2), ENCARG(quot)) ENCARG(cons_app(map, filter2)) -> c12(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c12(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c12(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c12(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c12(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c12(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c12(APP(false, filter2), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(filter2, filter2)) -> c12(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(quot, filter2)) -> c12(APP(quot, filter2), ENCARG(quot)) ENCARG(cons_app(0, filter2)) -> c12(APP(0, filter2), ENCARG(0)) ENCARG(cons_app(s, filter2)) -> c12(APP(s, filter2), ENCARG(s)) ENCARG(cons_app(nil, filter2)) -> c12(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(map, filter2)) -> c12(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(filter, filter2)) -> c12(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(true, filter2)) -> c12(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c12(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(minus, filter2)) -> c12(APP(minus, filter2), ENCARG(minus)) ENCARG(cons_app(cons, filter2)) -> c12(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(pred, filter2)) -> c12(APP(pred, filter2), ENCARG(pred)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) by ENCARG(cons_app(pred, true)) -> c12(APP(pred, true), ENCARG(pred)) ENCARG(cons_app(s, true)) -> c12(APP(s, true), ENCARG(s)) ENCARG(cons_app(minus, true)) -> c12(APP(minus, true), ENCARG(minus)) ENCARG(cons_app(0, true)) -> c12(APP(0, true), ENCARG(0)) ENCARG(cons_app(quot, true)) -> c12(APP(quot, true), ENCARG(quot)) ENCARG(cons_app(map, true)) -> c12(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c12(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c12(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c12(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c12(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c12(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c12(APP(false, true), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), true)) -> c12(APP(app(encArg(z0), encArg(z1)), true), ENCARG(cons_app(z0, z1))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, true)) -> c12(APP(pred, true), ENCARG(pred)) ENCARG(cons_app(s, true)) -> c12(APP(s, true), ENCARG(s)) ENCARG(cons_app(minus, true)) -> c12(APP(minus, true), ENCARG(minus)) ENCARG(cons_app(0, true)) -> c12(APP(0, true), ENCARG(0)) ENCARG(cons_app(quot, true)) -> c12(APP(quot, true), ENCARG(quot)) ENCARG(cons_app(map, true)) -> c12(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c12(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c12(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c12(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c12(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c12(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c12(APP(false, true), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(cons, true)) -> c12(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(0, true)) -> c12(APP(0, true), ENCARG(0)) ENCARG(cons_app(quot, true)) -> c12(APP(quot, true), ENCARG(quot)) ENCARG(cons_app(filter2, true)) -> c12(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(minus, true)) -> c12(APP(minus, true), ENCARG(minus)) ENCARG(cons_app(s, true)) -> c12(APP(s, true), ENCARG(s)) ENCARG(cons_app(nil, true)) -> c12(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(true, true)) -> c12(APP(true, true), ENCARG(true)) ENCARG(cons_app(map, true)) -> c12(APP(map, true), ENCARG(map)) ENCARG(cons_app(filter, true)) -> c12(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(false, true)) -> c12(APP(false, true), ENCARG(false)) ENCARG(cons_app(pred, true)) -> c12(APP(pred, true), ENCARG(pred)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) by ENCARG(cons_app(pred, false)) -> c12(APP(pred, false), ENCARG(pred)) ENCARG(cons_app(s, false)) -> c12(APP(s, false), ENCARG(s)) ENCARG(cons_app(minus, false)) -> c12(APP(minus, false), ENCARG(minus)) ENCARG(cons_app(0, false)) -> c12(APP(0, false), ENCARG(0)) ENCARG(cons_app(quot, false)) -> c12(APP(quot, false), ENCARG(quot)) ENCARG(cons_app(map, false)) -> c12(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c12(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c12(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c12(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c12(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c12(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c12(APP(false, false), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), false)) -> c12(APP(app(encArg(z0), encArg(z1)), false), ENCARG(cons_app(z0, z1))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, false)) -> c12(APP(pred, false), ENCARG(pred)) ENCARG(cons_app(s, false)) -> c12(APP(s, false), ENCARG(s)) ENCARG(cons_app(minus, false)) -> c12(APP(minus, false), ENCARG(minus)) ENCARG(cons_app(0, false)) -> c12(APP(0, false), ENCARG(0)) ENCARG(cons_app(quot, false)) -> c12(APP(quot, false), ENCARG(quot)) ENCARG(cons_app(map, false)) -> c12(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c12(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c12(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c12(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c12(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c12(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c12(APP(false, false), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (119) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(filter, false)) -> c12(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(cons, false)) -> c12(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(nil, false)) -> c12(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(0, false)) -> c12(APP(0, false), ENCARG(0)) ENCARG(cons_app(filter2, false)) -> c12(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c12(APP(true, false), ENCARG(true)) ENCARG(cons_app(pred, false)) -> c12(APP(pred, false), ENCARG(pred)) ENCARG(cons_app(false, false)) -> c12(APP(false, false), ENCARG(false)) ENCARG(cons_app(s, false)) -> c12(APP(s, false), ENCARG(s)) ENCARG(cons_app(map, false)) -> c12(APP(map, false), ENCARG(map)) ENCARG(cons_app(minus, false)) -> c12(APP(minus, false), ENCARG(minus)) ENCARG(cons_app(quot, false)) -> c12(APP(quot, false), ENCARG(quot)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_2, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(pred, x1)) -> c12(APP(pred, encArg(x1)), ENCARG(x1)) by ENCARG(cons_app(pred, pred)) -> c12(APP(pred, pred), ENCARG(pred)) ENCARG(cons_app(pred, s)) -> c12(APP(pred, s), ENCARG(s)) ENCARG(cons_app(pred, minus)) -> c12(APP(pred, minus), ENCARG(minus)) ENCARG(cons_app(pred, 0)) -> c12(APP(pred, 0), ENCARG(0)) ENCARG(cons_app(pred, quot)) -> c12(APP(pred, quot), ENCARG(quot)) ENCARG(cons_app(pred, map)) -> c12(APP(pred, map), ENCARG(map)) ENCARG(cons_app(pred, nil)) -> c12(APP(pred, nil), ENCARG(nil)) ENCARG(cons_app(pred, cons)) -> c12(APP(pred, cons), ENCARG(cons)) ENCARG(cons_app(pred, filter)) -> c12(APP(pred, filter), ENCARG(filter)) ENCARG(cons_app(pred, filter2)) -> c12(APP(pred, filter2), ENCARG(filter2)) ENCARG(cons_app(pred, true)) -> c12(APP(pred, true), ENCARG(true)) ENCARG(cons_app(pred, false)) -> c12(APP(pred, false), ENCARG(false)) ENCARG(cons_app(pred, cons_app(z0, z1))) -> c12(APP(pred, app(encArg(z0), encArg(z1))), ENCARG(cons_app(z0, z1))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(pred, pred)) -> c12(APP(pred, pred), ENCARG(pred)) ENCARG(cons_app(pred, s)) -> c12(APP(pred, s), ENCARG(s)) ENCARG(cons_app(pred, minus)) -> c12(APP(pred, minus), ENCARG(minus)) ENCARG(cons_app(pred, 0)) -> c12(APP(pred, 0), ENCARG(0)) ENCARG(cons_app(pred, quot)) -> c12(APP(pred, quot), ENCARG(quot)) ENCARG(cons_app(pred, map)) -> c12(APP(pred, map), ENCARG(map)) ENCARG(cons_app(pred, nil)) -> c12(APP(pred, nil), ENCARG(nil)) ENCARG(cons_app(pred, cons)) -> c12(APP(pred, cons), ENCARG(cons)) ENCARG(cons_app(pred, filter)) -> c12(APP(pred, filter), ENCARG(filter)) ENCARG(cons_app(pred, filter2)) -> c12(APP(pred, filter2), ENCARG(filter2)) ENCARG(cons_app(pred, true)) -> c12(APP(pred, true), ENCARG(true)) ENCARG(cons_app(pred, false)) -> c12(APP(pred, false), ENCARG(false)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2 ---------------------------------------- (123) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(pred, filter2)) -> c12(APP(pred, filter2), ENCARG(filter2)) ENCARG(cons_app(pred, quot)) -> c12(APP(pred, quot), ENCARG(quot)) ENCARG(cons_app(pred, pred)) -> c12(APP(pred, pred), ENCARG(pred)) ENCARG(cons_app(pred, nil)) -> c12(APP(pred, nil), ENCARG(nil)) ENCARG(cons_app(pred, true)) -> c12(APP(pred, true), ENCARG(true)) ENCARG(cons_app(pred, false)) -> c12(APP(pred, false), ENCARG(false)) ENCARG(cons_app(pred, filter)) -> c12(APP(pred, filter), ENCARG(filter)) ENCARG(cons_app(pred, cons)) -> c12(APP(pred, cons), ENCARG(cons)) ENCARG(cons_app(pred, 0)) -> c12(APP(pred, 0), ENCARG(0)) ENCARG(cons_app(pred, s)) -> c12(APP(pred, s), ENCARG(s)) ENCARG(cons_app(pred, minus)) -> c12(APP(pred, minus), ENCARG(minus)) ENCARG(cons_app(pred, map)) -> c12(APP(pred, map), ENCARG(map)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c28_2, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(minus, z0), app(s, app(s, z1))) -> c28(APP(pred, app(pred, app(app(minus, z0), z1))), APP(app(minus, z0), app(s, z1))) by APP(app(minus, z0), app(s, app(s, 0))) -> c28(APP(pred, app(pred, z0)), APP(app(minus, z0), app(s, 0))) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, 0))) -> c28(APP(pred, app(pred, z0)), APP(app(minus, z0), app(s, 0))) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, 0))) -> c28(APP(pred, app(pred, z0)), APP(app(minus, z0), app(s, 0))) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2 ---------------------------------------- (127) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(quot, app(s, z0)), app(s, app(s, z1))) -> c30(APP(app(quot, app(pred, app(app(minus, z0), z1))), app(s, app(s, z1))), APP(app(minus, z0), app(s, z1))) by APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2), APP(app(minus, z0), app(s, z1))) by APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(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)) -> c34(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)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, 0), app(quot, 0)), app(s, z0)), x2)), app(filter, app(quot, 0))), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1)))), app(quot, app(s, z0))), app(s, z1)), x2)), app(filter, app(quot, app(s, z0)))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, 0), app(quot, 0)), app(s, z0)), x2)), app(filter, app(quot, 0))), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1)))), app(quot, app(s, z0))), app(s, z1)), x2)), app(filter, app(quot, app(s, z0)))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, 0), app(quot, 0)), app(s, z0)), x2)), app(filter, app(quot, 0))), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(s, app(app(quot, app(app(minus, z0), z1)), app(s, z1)))), app(quot, app(s, z0))), app(s, z1)), x2)), app(filter, app(quot, app(s, z0)))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c34(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, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, pred) -> c(APP(encArg(x0), pred)) by ENCODE_APP(pred, pred) -> c(APP(pred, pred)) ENCODE_APP(s, pred) -> c(APP(s, pred)) ENCODE_APP(minus, pred) -> c(APP(minus, pred)) ENCODE_APP(0, pred) -> c(APP(0, pred)) ENCODE_APP(quot, pred) -> c(APP(quot, pred)) ENCODE_APP(map, pred) -> c(APP(map, pred)) ENCODE_APP(nil, pred) -> c(APP(nil, pred)) ENCODE_APP(cons, pred) -> c(APP(cons, pred)) ENCODE_APP(filter, pred) -> c(APP(filter, pred)) ENCODE_APP(filter2, pred) -> c(APP(filter2, pred)) ENCODE_APP(true, pred) -> c(APP(true, pred)) ENCODE_APP(false, pred) -> c(APP(false, pred)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(pred, pred) -> c(APP(pred, pred)) ENCODE_APP(s, pred) -> c(APP(s, pred)) ENCODE_APP(minus, pred) -> c(APP(minus, pred)) ENCODE_APP(0, pred) -> c(APP(0, pred)) ENCODE_APP(quot, pred) -> c(APP(quot, pred)) ENCODE_APP(map, pred) -> c(APP(map, pred)) ENCODE_APP(nil, pred) -> c(APP(nil, pred)) ENCODE_APP(cons, pred) -> c(APP(cons, pred)) ENCODE_APP(filter, pred) -> c(APP(filter, pred)) ENCODE_APP(filter2, pred) -> c(APP(filter2, pred)) ENCODE_APP(true, pred) -> c(APP(true, pred)) ENCODE_APP(false, pred) -> c(APP(false, pred)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCODE_APP(nil, pred) -> c(APP(nil, pred)) ENCODE_APP(pred, pred) -> c(APP(pred, pred)) ENCODE_APP(true, pred) -> c(APP(true, pred)) ENCODE_APP(s, pred) -> c(APP(s, pred)) ENCODE_APP(filter, pred) -> c(APP(filter, pred)) ENCODE_APP(cons, pred) -> c(APP(cons, pred)) ENCODE_APP(0, pred) -> c(APP(0, pred)) ENCODE_APP(false, pred) -> c(APP(false, pred)) ENCODE_APP(filter2, pred) -> c(APP(filter2, pred)) ENCODE_APP(quot, pred) -> c(APP(quot, pred)) ENCODE_APP(minus, pred) -> c(APP(minus, pred)) ENCODE_APP(map, pred) -> c(APP(map, pred)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) by ENCODE_APP(pred, s) -> c(APP(pred, s)) ENCODE_APP(s, s) -> c(APP(s, s)) ENCODE_APP(minus, s) -> c(APP(minus, s)) ENCODE_APP(0, s) -> c(APP(0, s)) ENCODE_APP(quot, s) -> c(APP(quot, s)) ENCODE_APP(map, s) -> c(APP(map, s)) ENCODE_APP(nil, s) -> c(APP(nil, s)) ENCODE_APP(cons, s) -> c(APP(cons, s)) ENCODE_APP(filter, s) -> c(APP(filter, s)) ENCODE_APP(filter2, s) -> c(APP(filter2, s)) ENCODE_APP(true, s) -> c(APP(true, s)) ENCODE_APP(false, s) -> c(APP(false, s)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(pred, s) -> c(APP(pred, s)) ENCODE_APP(s, s) -> c(APP(s, s)) ENCODE_APP(minus, s) -> c(APP(minus, s)) ENCODE_APP(0, s) -> c(APP(0, s)) ENCODE_APP(quot, s) -> c(APP(quot, s)) ENCODE_APP(map, s) -> c(APP(map, s)) ENCODE_APP(nil, s) -> c(APP(nil, s)) ENCODE_APP(cons, s) -> c(APP(cons, s)) ENCODE_APP(filter, s) -> c(APP(filter, s)) ENCODE_APP(filter2, s) -> c(APP(filter2, s)) ENCODE_APP(true, s) -> c(APP(true, s)) ENCODE_APP(false, s) -> c(APP(false, s)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCODE_APP(false, s) -> c(APP(false, s)) ENCODE_APP(filter, s) -> c(APP(filter, s)) ENCODE_APP(true, s) -> c(APP(true, s)) ENCODE_APP(s, s) -> c(APP(s, s)) ENCODE_APP(pred, s) -> c(APP(pred, s)) ENCODE_APP(0, s) -> c(APP(0, s)) ENCODE_APP(nil, s) -> c(APP(nil, s)) ENCODE_APP(minus, s) -> c(APP(minus, s)) ENCODE_APP(cons, s) -> c(APP(cons, s)) ENCODE_APP(filter2, s) -> c(APP(filter2, s)) ENCODE_APP(map, s) -> c(APP(map, s)) ENCODE_APP(quot, s) -> c(APP(quot, s)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (149) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) by ENCODE_APP(pred, minus) -> c(APP(pred, minus)) ENCODE_APP(s, minus) -> c(APP(s, minus)) ENCODE_APP(minus, minus) -> c(APP(minus, minus)) ENCODE_APP(0, minus) -> c(APP(0, minus)) ENCODE_APP(quot, minus) -> c(APP(quot, minus)) ENCODE_APP(map, minus) -> c(APP(map, minus)) ENCODE_APP(nil, minus) -> c(APP(nil, minus)) ENCODE_APP(cons, minus) -> c(APP(cons, minus)) ENCODE_APP(filter, minus) -> c(APP(filter, minus)) ENCODE_APP(filter2, minus) -> c(APP(filter2, minus)) ENCODE_APP(true, minus) -> c(APP(true, minus)) ENCODE_APP(false, minus) -> c(APP(false, minus)) ENCODE_APP(cons_app(z0, z1), minus) -> c(APP(app(encArg(z0), encArg(z1)), minus)) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) ENCODE_APP(pred, minus) -> c(APP(pred, minus)) ENCODE_APP(s, minus) -> c(APP(s, minus)) ENCODE_APP(minus, minus) -> c(APP(minus, minus)) ENCODE_APP(0, minus) -> c(APP(0, minus)) ENCODE_APP(quot, minus) -> c(APP(quot, minus)) ENCODE_APP(map, minus) -> c(APP(map, minus)) ENCODE_APP(nil, minus) -> c(APP(nil, minus)) ENCODE_APP(cons, minus) -> c(APP(cons, minus)) ENCODE_APP(filter, minus) -> c(APP(filter, minus)) ENCODE_APP(filter2, minus) -> c(APP(filter2, minus)) ENCODE_APP(true, minus) -> c(APP(true, minus)) ENCODE_APP(false, minus) -> c(APP(false, minus)) ENCODE_APP(cons_app(z0, z1), minus) -> c(APP(app(encArg(z0), encArg(z1)), minus)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCODE_APP(pred, minus) -> c(APP(pred, minus)) ENCODE_APP(filter2, minus) -> c(APP(filter2, minus)) ENCODE_APP(minus, minus) -> c(APP(minus, minus)) ENCODE_APP(cons, minus) -> c(APP(cons, minus)) ENCODE_APP(nil, minus) -> c(APP(nil, minus)) ENCODE_APP(map, minus) -> c(APP(map, minus)) ENCODE_APP(0, minus) -> c(APP(0, minus)) ENCODE_APP(false, minus) -> c(APP(false, minus)) ENCODE_APP(true, minus) -> c(APP(true, minus)) ENCODE_APP(filter, minus) -> c(APP(filter, minus)) ENCODE_APP(quot, minus) -> c(APP(quot, minus)) ENCODE_APP(s, minus) -> c(APP(s, minus)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) ENCODE_APP(cons_app(z0, z1), minus) -> c(APP(app(encArg(z0), encArg(z1)), minus)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) by ENCODE_APP(pred, 0) -> c(APP(pred, 0)) ENCODE_APP(s, 0) -> c(APP(s, 0)) ENCODE_APP(minus, 0) -> c(APP(minus, 0)) ENCODE_APP(0, 0) -> c(APP(0, 0)) ENCODE_APP(quot, 0) -> c(APP(quot, 0)) ENCODE_APP(map, 0) -> c(APP(map, 0)) ENCODE_APP(nil, 0) -> c(APP(nil, 0)) ENCODE_APP(cons, 0) -> c(APP(cons, 0)) ENCODE_APP(filter, 0) -> c(APP(filter, 0)) ENCODE_APP(filter2, 0) -> c(APP(filter2, 0)) ENCODE_APP(true, 0) -> c(APP(true, 0)) ENCODE_APP(false, 0) -> c(APP(false, 0)) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) ENCODE_APP(cons_app(z0, z1), minus) -> c(APP(app(encArg(z0), encArg(z1)), minus)) ENCODE_APP(pred, 0) -> c(APP(pred, 0)) ENCODE_APP(s, 0) -> c(APP(s, 0)) ENCODE_APP(minus, 0) -> c(APP(minus, 0)) ENCODE_APP(0, 0) -> c(APP(0, 0)) ENCODE_APP(quot, 0) -> c(APP(quot, 0)) ENCODE_APP(map, 0) -> c(APP(map, 0)) ENCODE_APP(nil, 0) -> c(APP(nil, 0)) ENCODE_APP(cons, 0) -> c(APP(cons, 0)) ENCODE_APP(filter, 0) -> c(APP(filter, 0)) ENCODE_APP(filter2, 0) -> c(APP(filter2, 0)) ENCODE_APP(true, 0) -> c(APP(true, 0)) ENCODE_APP(false, 0) -> c(APP(false, 0)) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (155) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCODE_APP(true, 0) -> c(APP(true, 0)) ENCODE_APP(false, 0) -> c(APP(false, 0)) ENCODE_APP(filter, 0) -> c(APP(filter, 0)) ENCODE_APP(minus, 0) -> c(APP(minus, 0)) ENCODE_APP(filter2, 0) -> c(APP(filter2, 0)) ENCODE_APP(cons, 0) -> c(APP(cons, 0)) ENCODE_APP(0, 0) -> c(APP(0, 0)) ENCODE_APP(nil, 0) -> c(APP(nil, 0)) ENCODE_APP(quot, 0) -> c(APP(quot, 0)) ENCODE_APP(pred, 0) -> c(APP(pred, 0)) ENCODE_APP(map, 0) -> c(APP(map, 0)) ENCODE_APP(s, 0) -> c(APP(s, 0)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) 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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) ENCODE_APP(cons_app(z0, z1), minus) -> c(APP(app(encArg(z0), encArg(z1)), minus)) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, quot) -> c(APP(encArg(x0), quot)) by ENCODE_APP(pred, quot) -> c(APP(pred, quot)) ENCODE_APP(s, quot) -> c(APP(s, quot)) ENCODE_APP(minus, quot) -> c(APP(minus, quot)) ENCODE_APP(0, quot) -> c(APP(0, quot)) ENCODE_APP(quot, quot) -> c(APP(quot, quot)) ENCODE_APP(map, quot) -> c(APP(map, quot)) ENCODE_APP(nil, quot) -> c(APP(nil, quot)) ENCODE_APP(cons, quot) -> c(APP(cons, quot)) ENCODE_APP(filter, quot) -> c(APP(filter, quot)) ENCODE_APP(filter2, quot) -> c(APP(filter2, quot)) ENCODE_APP(true, quot) -> c(APP(true, quot)) ENCODE_APP(false, quot) -> c(APP(false, quot)) ENCODE_APP(cons_app(z0, z1), quot) -> c(APP(app(encArg(z0), encArg(z1)), quot)) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) ENCODE_APP(cons_app(z0, z1), minus) -> c(APP(app(encArg(z0), encArg(z1)), minus)) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(pred, quot) -> c(APP(pred, quot)) ENCODE_APP(s, quot) -> c(APP(s, quot)) ENCODE_APP(minus, quot) -> c(APP(minus, quot)) ENCODE_APP(0, quot) -> c(APP(0, quot)) ENCODE_APP(quot, quot) -> c(APP(quot, quot)) ENCODE_APP(map, quot) -> c(APP(map, quot)) ENCODE_APP(nil, quot) -> c(APP(nil, quot)) ENCODE_APP(cons, quot) -> c(APP(cons, quot)) ENCODE_APP(filter, quot) -> c(APP(filter, quot)) ENCODE_APP(filter2, quot) -> c(APP(filter2, quot)) ENCODE_APP(true, quot) -> c(APP(true, quot)) ENCODE_APP(false, quot) -> c(APP(false, quot)) ENCODE_APP(cons_app(z0, z1), quot) -> c(APP(app(encArg(z0), encArg(z1)), quot)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCODE_APP(cons, quot) -> c(APP(cons, quot)) ENCODE_APP(s, quot) -> c(APP(s, quot)) ENCODE_APP(nil, quot) -> c(APP(nil, quot)) ENCODE_APP(minus, quot) -> c(APP(minus, quot)) ENCODE_APP(filter, quot) -> c(APP(filter, quot)) ENCODE_APP(map, quot) -> c(APP(map, quot)) ENCODE_APP(pred, quot) -> c(APP(pred, quot)) ENCODE_APP(true, quot) -> c(APP(true, quot)) ENCODE_APP(false, quot) -> c(APP(false, quot)) ENCODE_APP(filter2, quot) -> c(APP(filter2, quot)) ENCODE_APP(quot, quot) -> c(APP(quot, quot)) ENCODE_APP(0, quot) -> c(APP(0, quot)) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) ENCODE_APP(cons_app(z0, z1), minus) -> c(APP(app(encArg(z0), encArg(z1)), minus)) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_app(z0, z1), quot) -> c(APP(app(encArg(z0), encArg(z1)), quot)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) by ENCODE_APP(pred, map) -> c(APP(pred, map)) ENCODE_APP(s, map) -> c(APP(s, map)) ENCODE_APP(minus, map) -> c(APP(minus, map)) ENCODE_APP(0, map) -> c(APP(0, map)) ENCODE_APP(quot, map) -> c(APP(quot, 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)) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) ENCODE_APP(cons_app(z0, z1), minus) -> c(APP(app(encArg(z0), encArg(z1)), minus)) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_app(z0, z1), quot) -> c(APP(app(encArg(z0), encArg(z1)), quot)) ENCODE_APP(pred, map) -> c(APP(pred, map)) ENCODE_APP(s, map) -> c(APP(s, map)) ENCODE_APP(minus, map) -> c(APP(minus, map)) ENCODE_APP(0, map) -> c(APP(0, map)) ENCODE_APP(quot, map) -> c(APP(quot, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1 ---------------------------------------- (163) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCODE_APP(true, map) -> c(APP(true, map)) ENCODE_APP(s, map) -> c(APP(s, map)) ENCODE_APP(pred, map) -> c(APP(pred, map)) ENCODE_APP(cons, map) -> c(APP(cons, map)) ENCODE_APP(0, map) -> c(APP(0, map)) ENCODE_APP(filter2, map) -> c(APP(filter2, map)) ENCODE_APP(filter, map) -> c(APP(filter, map)) ENCODE_APP(nil, map) -> c(APP(nil, map)) ENCODE_APP(minus, map) -> c(APP(minus, map)) ENCODE_APP(quot, map) -> c(APP(quot, map)) ENCODE_APP(map, map) -> c(APP(map, map)) ENCODE_APP(false, map) -> c(APP(false, map)) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: encArg(pred) -> pred encArg(s) -> s encArg(minus) -> minus encArg(0) -> 0 encArg(quot) -> quot 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(pred, app(s, z0)) -> z0 app(app(minus, z0), 0) -> z0 app(app(minus, z0), app(s, z1)) -> app(pred, app(app(minus, z0), z1)) app(app(quot, 0), app(s, z0)) -> 0 app(app(quot, app(s, z0)), app(s, z1)) -> app(s, app(app(quot, app(app(minus, z0), z1)), app(s, 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(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(quot, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(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(pred, x1) -> c(APP(pred, 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, pred))) -> c12(APP(encArg(x0), app(encArg(x1), pred)), ENCARG(x0), ENCARG(cons_app(x1, pred))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, quot))) -> c12(APP(encArg(x0), app(encArg(x1), quot)), ENCARG(x0), ENCARG(cons_app(x1, quot))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(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)))) -> c12(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(pred, x2))) -> c12(APP(encArg(x0), app(pred, encArg(x2))), ENCARG(x0), ENCARG(cons_app(pred, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(quot, x2))) -> c12(APP(encArg(x0), app(quot, encArg(x2))), ENCARG(x0), ENCARG(cons_app(quot, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(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))) -> c12(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))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(pred, cons_app(x1, x2))) -> c12(APP(pred, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(quot, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, pred), x2)) -> c12(APP(app(encArg(x0), pred), encArg(x2)), ENCARG(cons_app(x0, pred)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, quot), x2)) -> c12(APP(app(encArg(x0), quot), encArg(x2)), ENCARG(cons_app(x0, quot)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(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)) -> c12(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(pred, x1), x2)) -> c12(APP(app(pred, encArg(x1)), encArg(x2)), ENCARG(cons_app(pred, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(quot, x1), x2)) -> c12(APP(app(quot, encArg(x1)), encArg(x2)), ENCARG(cons_app(quot, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(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)) -> c12(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), pred)) -> c12(APP(app(encArg(x0), encArg(x1)), pred), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), quot)) -> c12(APP(app(encArg(x0), encArg(x1)), quot), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), pred) -> c(APP(app(encArg(z0), encArg(z1)), pred)) ENCODE_APP(cons_app(z0, z1), s) -> c(APP(app(encArg(z0), encArg(z1)), s)) ENCODE_APP(cons_app(z0, z1), minus) -> c(APP(app(encArg(z0), encArg(z1)), minus)) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_app(z0, z1), quot) -> c(APP(app(encArg(z0), encArg(z1)), quot)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) S tuples: APP(pred, app(s, z0)) -> c26 APP(app(minus, z0), 0) -> c27 APP(app(quot, 0), app(s, z0)) -> c29 APP(app(map, z0), nil) -> c31 APP(app(filter, z0), nil) -> c33 APP(app(map, z0), app(app(cons, z1), z2)) -> c32(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c35(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(minus, z0), app(s, 0)) -> c1(APP(pred, z0)) APP(app(minus, z0), app(s, 0)) -> c1(APP(app(minus, z0), 0)) APP(app(quot, app(s, z0)), app(s, 0)) -> c30(APP(app(quot, z0), app(s, 0)), APP(app(minus, z0), 0)) APP(app(filter, pred), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(app(app(filter2, z0), pred), app(s, z0)), x2), APP(pred, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c34(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2)) -> c34(APP(app(quot, 0), app(s, z0))) APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c34(APP(app(quot, app(s, z0)), app(s, z1))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c34(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c34(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c34(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c34(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(minus, z0), app(s, app(s, app(s, z1)))) -> c28(APP(pred, app(pred, app(pred, app(app(minus, z0), z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(pred, app(pred, z0))) APP(app(minus, z0), app(s, app(s, 0))) -> c2(APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, 0))) -> c30(APP(app(quot, app(pred, z0)), app(s, app(s, 0))), APP(app(minus, z0), app(s, 0))) APP(app(quot, app(s, z0)), app(s, app(s, app(s, z1)))) -> c30(APP(app(quot, app(pred, app(pred, app(app(minus, z0), z1)))), app(s, app(s, app(s, z1)))), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, 0)), x2)) -> c34(APP(app(app(app(filter2, app(pred, z0)), app(minus, z0)), app(s, 0)), x2), APP(app(minus, z0), app(s, 0))) APP(app(filter, app(minus, z0)), app(app(cons, app(s, app(s, z1))), x2)) -> c34(APP(app(app(app(filter2, app(pred, app(pred, app(app(minus, z0), z1)))), app(minus, z0)), app(s, app(s, z1))), x2), APP(app(minus, z0), app(s, app(s, z1)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c34(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, pred)), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), pred), app(s, z0)), x2)), app(filter, pred)), app(app(cons, app(s, z0)), x2)), x3), APP(app(filter, pred), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, z0), app(minus, z0)), 0), x2)), app(filter, app(minus, z0))), app(app(cons, 0), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(minus, z0))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(app(app(filter2, app(app(app(app(filter2, app(pred, app(app(minus, z0), z1))), app(minus, z0)), app(s, z1)), x2)), app(filter, app(minus, z0))), app(app(cons, app(s, z1)), x2)), x3), APP(app(filter, app(minus, z0)), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c34(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)) -> c34(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(quot, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c34(APP(app(filter, app(quot, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(quot, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c34(APP(app(filter, app(quot, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(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)) -> c34(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c34(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: c26, c27, c29, c31, c33, c32_2, c35_1, c36_1, c12_1, c1_1, c30_2, c34_2, c34_1, c_1, c12_3, c12_2, c28_2, c2_1