/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 203 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 8 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 8738 ms] (14) BOUNDS(1, INF) (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) NarrowingProof [BOTH BOUNDS(ID, ID), 16 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), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 15 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 3 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 41 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 101 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 18 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 18 ms] (68) CdtProblem (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (84) CdtProblem (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (88) CdtProblem (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 6 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem ---------------------------------------- (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(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) app(app(:, app(app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) app(app(:, z), app(app(+, x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) 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(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a 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(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) app(app(:, app(app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) app(app(:, z), app(app(+, x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) 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(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a 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(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) app(app(:, app(app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) app(app(:, z), app(app(+, x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) 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(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) app(app(:, app(app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) app(app(:, z), app(app(+, x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) 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(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) app(app(:, app(app(+', x), y)), z) -> app(app(+', app(app(:, x), z)), app(app(:, y), z)) app(app(:, z), app(app(+', x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+', x), a)) 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(:) -> : encArg(+') -> +' encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> +' encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: app(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) app(app(:, app(app(+', x), y)), z) -> app(app(+', app(app(:, x), z)), app(app(:, y), z)) app(app(:, z), app(app(+', x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+', x), a)) 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(:) -> : encArg(+') -> +' encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> +' encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app : :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app +' :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app g :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app a :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app map :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app nil :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app cons :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app filter :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app true :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app false :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encArg :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_: :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_+ :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_g :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_a :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app hole_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app1_0 :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0 :: Nat -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (12) Obligation: Innermost TRS: Rules: app(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) app(app(:, app(app(+', x), y)), z) -> app(app(+', app(app(:, x), z)), app(app(:, y), z)) app(app(:, z), app(app(+', x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+', x), a)) 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(:) -> : encArg(+') -> +' encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> +' encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app : :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app +' :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app g :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app a :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app map :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app nil :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app cons :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app filter :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app true :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app false :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encArg :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_: :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_+ :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_g :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_a :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app hole_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app1_0 :: ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0 :: Nat -> ::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app Generator Equations: gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0(0) <=> : gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0(+(x, 1)) <=> cons_app(:, gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0(x)) The following defined symbols remain to be analysed: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0(n2844_0)) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0(0)) Induction Step: encArg(gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0(+(n2844_0, 1))) ->_R^Omega(0) app(encArg(:), encArg(gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0(n2844_0))) ->_R^Omega(0) app(:, encArg(gen_::+':g:a:map:nil:cons:filter:filter2:true:false:cons_app2_0(n2844_0))) ->_IH app(:, *3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (14) BOUNDS(1, INF) ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) app(app(:, app(app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) app(app(:, z), app(app(+, x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) 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(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a 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(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(:, c_app(c_app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) app(c_app(:, c_app(c_app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) app(c_app(:, z), c_app(c_app(+, x), c_app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) The (relative) TRS S consists of the following rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a 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(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(:, c_app(c_app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) [1] app(c_app(:, c_app(c_app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) [1] app(c_app(:, z), c_app(c_app(+, x), c_app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) [1] encArg(:) -> : [0] encArg(+) -> + [0] encArg(g) -> g [0] encArg(a) -> a [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_: -> : [0] encode_+ -> + [0] encode_g -> g [0] encode_a -> a [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(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(:, c_app(c_app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) [1] app(c_app(:, c_app(c_app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) [1] app(c_app(:, z), c_app(c_app(+, x), c_app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) [1] encArg(:) -> : [0] encArg(+) -> + [0] encArg(g) -> g [0] encArg(a) -> a [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_: -> : [0] encode_+ -> + [0] encode_g -> g [0] encode_a -> a [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:filter2:true:filter:false:nil:::+:g:a:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app c_app :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app map :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app cons :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app filter2 :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app true :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app filter :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app false :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app nil :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app : :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app + :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encArg :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app cons_app :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_app :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_: :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_+ :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_map :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_true :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app encode_false :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a: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_: encode_+ encode_g encode_a 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_: -> null_encode_: [0] encode_+ -> null_encode_+ [0] encode_g -> null_encode_g [0] encode_a -> null_encode_a [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_:, null_encode_+, null_encode_g, null_encode_a, 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(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(:, c_app(c_app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) [1] app(c_app(:, c_app(c_app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) [1] app(c_app(:, z), c_app(c_app(+, x), c_app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) [1] encArg(:) -> : [0] encArg(+) -> + [0] encArg(g) -> g [0] encArg(a) -> a [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_: -> : [0] encode_+ -> + [0] encode_g -> g [0] encode_a -> a [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_: -> null_encode_: [0] encode_+ -> null_encode_+ [0] encode_g -> null_encode_g [0] encode_a -> null_encode_a [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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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_g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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_a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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(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, f6)), c_app(c_app(cons, c_app(c_app(cons, x7), xs1)), xs)) -> app(app(app(app(filter2, app(app(cons, app(f6, x7)), app(app(map, f6), xs1))), c_app(map, f6)), c_app(c_app(cons, x7), xs1)), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, true), f7), x8)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(cons, x8), app(app(filter, f7), x))), c_app(c_app(c_app(filter2, true), f7), x8)), x), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, false), f8), x9)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(filter, f8), x)), c_app(c_app(c_app(filter2, false), f8), x9)), x), xs) [2] app(c_app(filter, c_app(map, f9)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(map, f9)), nil), xs) [2] app(c_app(filter, c_app(filter, f10)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(filter, f10)), nil), xs) [2] app(c_app(filter, c_app(filter, f11)), c_app(c_app(cons, c_app(c_app(cons, x10), xs2)), xs)) -> app(app(app(app(filter2, app(app(app(app(filter2, app(f11, x10)), f11), x10), xs2)), c_app(filter, f11)), c_app(c_app(cons, x10), xs2)), xs) [2] app(c_app(filter, c_app(:, c_app(c_app(+, x11), y2))), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(+, app(app(:, x11), x)), app(app(:, y2), x))), c_app(:, c_app(c_app(+, x11), y2))), x), xs) [2] app(c_app(filter, c_app(:, c_app(c_app(:, x12), y3))), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(:, x12), app(app(:, y3), x))), c_app(:, c_app(c_app(:, x12), y3))), x), xs) [2] app(c_app(filter, c_app(:, z'')), c_app(c_app(cons, c_app(c_app(+, x13), c_app(f12, y4))), xs)) -> app(app(app(app(filter2, app(app(:, app(app(g, z''), y4)), app(app(+, x13), a))), c_app(:, z'')), c_app(c_app(+, x13), c_app(f12, 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(:, c_app(c_app(+, x), y)), z) -> app(app(+, app(c_app(:, x), z)), app(c_app(:, y), z)) [1] app(c_app(:, c_app(c_app(+, x), y)), z) -> app(app(+, app(c_app(:, x), z)), app(null_app, z)) [1] app(c_app(:, c_app(c_app(+, x), y)), z) -> app(app(+, app(null_app, z)), app(c_app(:, y), z)) [1] app(c_app(:, c_app(c_app(+, x), y)), z) -> app(app(+, app(null_app, z)), app(null_app, z)) [1] app(c_app(:, c_app(c_app(:, x), y)), z) -> app(c_app(:, x), app(c_app(:, y), z)) [1] app(c_app(:, c_app(c_app(:, x), y)), z) -> app(c_app(:, x), app(null_app, z)) [1] app(c_app(:, c_app(c_app(:, x), y)), z) -> app(null_app, app(c_app(:, y), z)) [1] app(c_app(:, c_app(c_app(:, x), y)), z) -> app(null_app, app(null_app, z)) [1] app(c_app(:, z), c_app(c_app(+, x), c_app(f, y))) -> app(app(:, app(c_app(g, z), y)), app(c_app(+, x), a)) [1] app(c_app(:, z), c_app(c_app(+, x), c_app(f, y))) -> app(app(:, app(c_app(g, z), y)), app(null_app, a)) [1] app(c_app(:, z), c_app(c_app(+, x), c_app(f, y))) -> app(app(:, app(null_app, y)), app(c_app(+, x), a)) [1] app(c_app(:, z), c_app(c_app(+, x), c_app(f, y))) -> app(app(:, app(null_app, y)), app(null_app, a)) [1] encArg(:) -> : [0] encArg(+) -> + [0] encArg(g) -> g [0] encArg(a) -> a [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_: -> : [0] encode_+ -> + [0] encode_g -> g [0] encode_a -> a [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_: -> null_encode_: [0] encode_+ -> null_encode_+ [0] encode_g -> null_encode_g [0] encode_a -> null_encode_a [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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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_g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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_a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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 => 8 cons => 3 filter2 => 6 true => 10 filter => 5 false => 4 nil => 9 : => 1 + => 0 g => 7 a => 2 null_encArg => 0 null_encode_app => 0 null_encode_: => 0 null_encode_+ => 0 null_encode_g => 0 null_encode_a => 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', z1) -{ 2 }-> app(app(app(app(6, app(app(app(app(6, app(f11, x10)), f11), x10), xs2)), 1 + 5 + f11), 1 + (1 + 3 + x10) + xs2), xs) :|: xs >= 0, z1 = 1 + (1 + 3 + (1 + (1 + 3 + x10) + xs2)) + xs, xs2 >= 0, z' = 1 + 5 + (1 + 5 + f11), f11 >= 0, x10 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(5, f8), x)), 1 + (1 + (1 + 6 + 4) + f8) + x9), x), xs) :|: z' = 1 + 5 + (1 + (1 + (1 + 6 + 4) + f8) + x9), f8 >= 0, xs >= 0, x >= 0, z1 = 1 + (1 + 3 + x) + xs, x9 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(3, x8), app(app(5, f7), x))), 1 + (1 + (1 + 6 + 10) + f7) + x8), x), xs) :|: xs >= 0, z' = 1 + 5 + (1 + (1 + (1 + 6 + 10) + f7) + x8), x8 >= 0, x >= 0, f7 >= 0, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(3, app(f6, x7)), app(app(8, f6), xs1))), 1 + 8 + f6), 1 + (1 + 3 + x7) + xs1), xs) :|: xs >= 0, z1 = 1 + (1 + 3 + (1 + (1 + 3 + x7) + xs1)) + xs, x7 >= 0, f6 >= 0, xs1 >= 0, z' = 1 + 5 + (1 + 8 + f6) app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(1, x12), app(app(1, y3), x))), 1 + 1 + (1 + (1 + 1 + x12) + y3)), x), xs) :|: xs >= 0, z' = 1 + 5 + (1 + 1 + (1 + (1 + 1 + x12) + y3)), x >= 0, x12 >= 0, y3 >= 0, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(1, app(app(7, z''), y4)), app(app(0, x13), 2))), 1 + 1 + z''), 1 + (1 + 0 + x13) + (1 + f12 + y4)), xs) :|: z' = 1 + 5 + (1 + 1 + z''), x13 >= 0, xs >= 0, f12 >= 0, z'' >= 0, z1 = 1 + (1 + 3 + (1 + (1 + 0 + x13) + (1 + f12 + y4))) + xs, y4 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(0, app(app(1, x11), x)), app(app(1, y2), x))), 1 + 1 + (1 + (1 + 0 + x11) + y2)), x), xs) :|: xs >= 0, x >= 0, z' = 1 + 5 + (1 + 1 + (1 + (1 + 0 + x11) + y2)), x11 >= 0, z1 = 1 + (1 + 3 + x) + xs, y2 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, 9), 1 + 8 + f9), 9), xs) :|: z' = 1 + 5 + (1 + 8 + f9), xs >= 0, z1 = 1 + (1 + 3 + 9) + xs, f9 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, 9), 1 + 5 + f10), 9), xs) :|: xs >= 0, z1 = 1 + (1 + 3 + 9) + xs, f10 >= 0, z' = 1 + 5 + (1 + 5 + f10) app(z', z1) -{ 1 }-> app(app(app(app(6, 0), f), x), xs) :|: f >= 0, xs >= 0, x >= 0, z' = 1 + 5 + f, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 1 }-> app(app(app(app(6, 1 + f + x), f), x), xs) :|: f >= 0, xs >= 0, x >= 0, z' = 1 + 5 + f, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 1 }-> app(app(3, app(f, x)), app(app(8, f), xs)) :|: f >= 0, xs >= 0, z' = 1 + 8 + f, x >= 0, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 1 }-> app(app(1, app(0, y)), app(0, 2)) :|: z' = 1 + 1 + z, f >= 0, z >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 0 + x) + (1 + f + y) app(z', z1) -{ 1 }-> app(app(1, app(0, y)), app(1 + 0 + x, 2)) :|: z' = 1 + 1 + z, f >= 0, z >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 0 + x) + (1 + f + y) app(z', z1) -{ 1 }-> app(app(1, app(1 + 7 + z, y)), app(0, 2)) :|: z' = 1 + 1 + z, f >= 0, z >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 0 + x) + (1 + f + y) app(z', z1) -{ 1 }-> app(app(1, app(1 + 7 + z, y)), app(1 + 0 + x, 2)) :|: z' = 1 + 1 + z, f >= 0, z >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 0 + x) + (1 + f + y) app(z', z1) -{ 1 }-> app(app(0, app(0, z)), app(0, z)) :|: z1 = z, z >= 0, z' = 1 + 1 + (1 + (1 + 0 + x) + y), x >= 0, y >= 0 app(z', z1) -{ 1 }-> app(app(0, app(0, z)), app(1 + 1 + y, z)) :|: z1 = z, z >= 0, z' = 1 + 1 + (1 + (1 + 0 + x) + y), x >= 0, y >= 0 app(z', z1) -{ 1 }-> app(app(0, app(1 + 1 + x, z)), app(0, z)) :|: z1 = z, z >= 0, z' = 1 + 1 + (1 + (1 + 0 + x) + y), x >= 0, y >= 0 app(z', z1) -{ 1 }-> app(app(0, app(1 + 1 + x, z)), app(1 + 1 + y, z)) :|: z1 = z, z >= 0, z' = 1 + 1 + (1 + (1 + 0 + x) + y), x >= 0, y >= 0 app(z', z1) -{ 1 }-> app(0, xs) :|: f >= 0, xs >= 0, z' = 1 + (1 + (1 + 6 + 4) + f) + x, x >= 0, z1 = xs app(z', z1) -{ 1 }-> app(0, app(0, xs)) :|: f >= 0, xs >= 0, z' = 1 + (1 + (1 + 6 + 10) + f) + x, x >= 0, z1 = xs app(z', z1) -{ 1 }-> app(0, app(0, z)) :|: z1 = z, z >= 0, x >= 0, y >= 0, z' = 1 + 1 + (1 + (1 + 1 + x) + y) app(z', z1) -{ 1 }-> app(0, app(1 + 5 + f, xs)) :|: f >= 0, xs >= 0, z' = 1 + (1 + (1 + 6 + 10) + f) + x, x >= 0, z1 = xs app(z', z1) -{ 1 }-> app(0, app(1 + 1 + y, z)) :|: z1 = z, z >= 0, x >= 0, y >= 0, z' = 1 + 1 + (1 + (1 + 1 + x) + y) app(z', z1) -{ 1 }-> app(1 + 5 + f, xs) :|: f >= 0, xs >= 0, z' = 1 + (1 + (1 + 6 + 4) + f) + x, x >= 0, z1 = xs app(z', z1) -{ 1 }-> app(1 + 3 + x, app(0, xs)) :|: f >= 0, xs >= 0, z' = 1 + (1 + (1 + 6 + 10) + f) + x, x >= 0, z1 = xs app(z', z1) -{ 1 }-> app(1 + 3 + x, app(1 + 5 + f, xs)) :|: f >= 0, xs >= 0, z' = 1 + (1 + (1 + 6 + 10) + f) + x, x >= 0, z1 = xs app(z', z1) -{ 1 }-> app(1 + 1 + x, app(0, z)) :|: z1 = z, z >= 0, x >= 0, y >= 0, z' = 1 + 1 + (1 + (1 + 1 + x) + y) app(z', z1) -{ 1 }-> app(1 + 1 + x, app(1 + 1 + y, z)) :|: z1 = z, z >= 0, x >= 0, y >= 0, z' = 1 + 1 + (1 + (1 + 1 + x) + y) app(z', z1) -{ 1 }-> 9 :|: f >= 0, z1 = 9, z' = 1 + 8 + f app(z', z1) -{ 1 }-> 9 :|: f >= 0, z1 = 9, z' = 1 + 5 + f app(z', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v1, v1 >= 0, z' = v0 app(z', z1) -{ 0 }-> 1 + x0 + x1 :|: x0 >= 0, x1 >= 0, z1 = x1, z' = x0 encArg(z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 9 :|: z' = 9 encArg(z') -{ 0 }-> 8 :|: z' = 8 encArg(z') -{ 0 }-> 7 :|: z' = 7 encArg(z') -{ 0 }-> 6 :|: z' = 6 encArg(z') -{ 0 }-> 5 :|: z' = 5 encArg(z') -{ 0 }-> 4 :|: z' = 4 encArg(z') -{ 0 }-> 3 :|: z' = 3 encArg(z') -{ 0 }-> 2 :|: z' = 2 encArg(z') -{ 0 }-> 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 :|: encode_: -{ 0 }-> 1 :|: encode_: -{ 0 }-> 0 :|: encode_a -{ 0 }-> 2 :|: encode_a -{ 0 }-> 0 :|: encode_app(z', z1) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z1 = x_2, z' = x_1, x_2 >= 0 encode_app(z', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v1, v1 >= 0, z' = v0 encode_cons -{ 0 }-> 3 :|: encode_cons -{ 0 }-> 0 :|: encode_false -{ 0 }-> 4 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 5 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 6 :|: encode_filter2 -{ 0 }-> 0 :|: encode_g -{ 0 }-> 7 :|: encode_g -{ 0 }-> 0 :|: encode_map -{ 0 }-> 8 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 9 :|: encode_nil -{ 0 }-> 0 :|: encode_true -{ 0 }-> 10 :|: 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', z1) -{ 2 }-> app(app(app(app(6, app(app(app(app(6, app(z' - 12, x10)), z' - 12), x10), xs2)), 1 + 5 + (z' - 12)), 1 + (1 + 3 + x10) + xs2), xs) :|: xs >= 0, z1 = 1 + (1 + 3 + (1 + (1 + 3 + x10) + xs2)) + xs, xs2 >= 0, z' - 12 >= 0, x10 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(5, f8), x)), 1 + (1 + (1 + 6 + 4) + f8) + x9), x), xs) :|: z' = 1 + 5 + (1 + (1 + (1 + 6 + 4) + f8) + x9), f8 >= 0, xs >= 0, x >= 0, z1 = 1 + (1 + 3 + x) + xs, x9 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(3, x8), app(app(5, f7), x))), 1 + (1 + (1 + 6 + 10) + f7) + x8), x), xs) :|: xs >= 0, z' = 1 + 5 + (1 + (1 + (1 + 6 + 10) + f7) + x8), x8 >= 0, x >= 0, f7 >= 0, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(3, app(z' - 15, x7)), app(app(8, z' - 15), xs1))), 1 + 8 + (z' - 15)), 1 + (1 + 3 + x7) + xs1), xs) :|: xs >= 0, z1 = 1 + (1 + 3 + (1 + (1 + 3 + x7) + xs1)) + xs, x7 >= 0, z' - 15 >= 0, xs1 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(1, x12), app(app(1, y3), x))), 1 + 1 + (1 + (1 + 1 + x12) + y3)), x), xs) :|: xs >= 0, z' = 1 + 5 + (1 + 1 + (1 + (1 + 1 + x12) + y3)), x >= 0, x12 >= 0, y3 >= 0, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(1, app(app(7, z' - 8), y4)), app(app(0, x13), 2))), 1 + 1 + (z' - 8)), 1 + (1 + 0 + x13) + (1 + f12 + y4)), xs) :|: x13 >= 0, xs >= 0, f12 >= 0, z' - 8 >= 0, z1 = 1 + (1 + 3 + (1 + (1 + 0 + x13) + (1 + f12 + y4))) + xs, y4 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, app(app(0, app(app(1, x11), x)), app(app(1, y2), x))), 1 + 1 + (1 + (1 + 0 + x11) + y2)), x), xs) :|: xs >= 0, x >= 0, z' = 1 + 5 + (1 + 1 + (1 + (1 + 0 + x11) + y2)), x11 >= 0, z1 = 1 + (1 + 3 + x) + xs, y2 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, 9), 1 + 8 + (z' - 15)), 9), z1 - 14) :|: z1 - 14 >= 0, z' - 15 >= 0 app(z', z1) -{ 2 }-> app(app(app(app(6, 9), 1 + 5 + (z' - 12)), 9), z1 - 14) :|: z1 - 14 >= 0, z' - 12 >= 0 app(z', z1) -{ 1 }-> app(app(app(app(6, 0), z' - 6), x), xs) :|: z' - 6 >= 0, xs >= 0, x >= 0, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 1 }-> app(app(app(app(6, 1 + (z' - 6) + x), z' - 6), x), xs) :|: z' - 6 >= 0, xs >= 0, x >= 0, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 1 }-> app(app(3, app(z' - 9, x)), app(app(8, z' - 9), xs)) :|: z' - 9 >= 0, xs >= 0, x >= 0, z1 = 1 + (1 + 3 + x) + xs app(z', z1) -{ 1 }-> app(app(1, app(0, y)), app(0, 2)) :|: f >= 0, z' - 2 >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 0 + x) + (1 + f + y) app(z', z1) -{ 1 }-> app(app(1, app(0, y)), app(1 + 0 + x, 2)) :|: f >= 0, z' - 2 >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 0 + x) + (1 + f + y) app(z', z1) -{ 1 }-> app(app(1, app(1 + 7 + (z' - 2), y)), app(0, 2)) :|: f >= 0, z' - 2 >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 0 + x) + (1 + f + y) app(z', z1) -{ 1 }-> app(app(1, app(1 + 7 + (z' - 2), y)), app(1 + 0 + x, 2)) :|: f >= 0, z' - 2 >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 0 + x) + (1 + f + y) app(z', z1) -{ 1 }-> app(app(0, app(0, z1)), app(0, z1)) :|: z1 >= 0, z' = 1 + 1 + (1 + (1 + 0 + x) + y), x >= 0, y >= 0 app(z', z1) -{ 1 }-> app(app(0, app(0, z1)), app(1 + 1 + y, z1)) :|: z1 >= 0, z' = 1 + 1 + (1 + (1 + 0 + x) + y), x >= 0, y >= 0 app(z', z1) -{ 1 }-> app(app(0, app(1 + 1 + x, z1)), app(0, z1)) :|: z1 >= 0, z' = 1 + 1 + (1 + (1 + 0 + x) + y), x >= 0, y >= 0 app(z', z1) -{ 1 }-> app(app(0, app(1 + 1 + x, z1)), app(1 + 1 + y, z1)) :|: z1 >= 0, z' = 1 + 1 + (1 + (1 + 0 + x) + y), x >= 0, y >= 0 app(z', z1) -{ 1 }-> app(0, z1) :|: f >= 0, z1 >= 0, z' = 1 + (1 + (1 + 6 + 4) + f) + x, x >= 0 app(z', z1) -{ 1 }-> app(0, app(0, z1)) :|: f >= 0, z1 >= 0, z' = 1 + (1 + (1 + 6 + 10) + f) + x, x >= 0 app(z', z1) -{ 1 }-> app(0, app(0, z1)) :|: z1 >= 0, x >= 0, y >= 0, z' = 1 + 1 + (1 + (1 + 1 + x) + y) app(z', z1) -{ 1 }-> app(0, app(1 + 5 + f, z1)) :|: f >= 0, z1 >= 0, z' = 1 + (1 + (1 + 6 + 10) + f) + x, x >= 0 app(z', z1) -{ 1 }-> app(0, app(1 + 1 + y, z1)) :|: z1 >= 0, x >= 0, y >= 0, z' = 1 + 1 + (1 + (1 + 1 + x) + y) app(z', z1) -{ 1 }-> app(1 + 5 + f, z1) :|: f >= 0, z1 >= 0, z' = 1 + (1 + (1 + 6 + 4) + f) + x, x >= 0 app(z', z1) -{ 1 }-> app(1 + 3 + x, app(0, z1)) :|: f >= 0, z1 >= 0, z' = 1 + (1 + (1 + 6 + 10) + f) + x, x >= 0 app(z', z1) -{ 1 }-> app(1 + 3 + x, app(1 + 5 + f, z1)) :|: f >= 0, z1 >= 0, z' = 1 + (1 + (1 + 6 + 10) + f) + x, x >= 0 app(z', z1) -{ 1 }-> app(1 + 1 + x, app(0, z1)) :|: z1 >= 0, x >= 0, y >= 0, z' = 1 + 1 + (1 + (1 + 1 + x) + y) app(z', z1) -{ 1 }-> app(1 + 1 + x, app(1 + 1 + y, z1)) :|: z1 >= 0, x >= 0, y >= 0, z' = 1 + 1 + (1 + (1 + 1 + x) + y) app(z', z1) -{ 1 }-> 9 :|: z' - 9 >= 0, z1 = 9 app(z', z1) -{ 1 }-> 9 :|: z' - 6 >= 0, z1 = 9 app(z', z1) -{ 0 }-> 0 :|: z' >= 0, z1 >= 0 app(z', z1) -{ 0 }-> 1 + z' + z1 :|: z' >= 0, z1 >= 0 encArg(z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 9 :|: z' = 9 encArg(z') -{ 0 }-> 8 :|: z' = 8 encArg(z') -{ 0 }-> 7 :|: z' = 7 encArg(z') -{ 0 }-> 6 :|: z' = 6 encArg(z') -{ 0 }-> 5 :|: z' = 5 encArg(z') -{ 0 }-> 4 :|: z' = 4 encArg(z') -{ 0 }-> 3 :|: z' = 3 encArg(z') -{ 0 }-> 2 :|: z' = 2 encArg(z') -{ 0 }-> 10 :|: z' = 10 encArg(z') -{ 0 }-> 1 :|: z' = 1 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encode_+ -{ 0 }-> 0 :|: encode_: -{ 0 }-> 1 :|: encode_: -{ 0 }-> 0 :|: encode_a -{ 0 }-> 2 :|: encode_a -{ 0 }-> 0 :|: encode_app(z', z1) -{ 0 }-> app(encArg(z'), encArg(z1)) :|: z' >= 0, z1 >= 0 encode_app(z', z1) -{ 0 }-> 0 :|: z' >= 0, z1 >= 0 encode_cons -{ 0 }-> 3 :|: encode_cons -{ 0 }-> 0 :|: encode_false -{ 0 }-> 4 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 5 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 6 :|: encode_filter2 -{ 0 }-> 0 :|: encode_g -{ 0 }-> 7 :|: encode_g -{ 0 }-> 0 :|: encode_map -{ 0 }-> 8 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 9 :|: encode_nil -{ 0 }-> 0 :|: encode_true -{ 0 }-> 10 :|: 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_: -> null_encode_: [0] encode_+ -> null_encode_+ [0] encode_g -> null_encode_g [0] encode_a -> null_encode_a [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_:, null_encode_+, null_encode_g, null_encode_a, 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(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(:, c_app(c_app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z)) [1] app(c_app(:, c_app(c_app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z)) [1] app(c_app(:, z), c_app(c_app(+, x), c_app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a)) [1] encArg(:) -> : [0] encArg(+) -> + [0] encArg(g) -> g [0] encArg(a) -> a [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_: -> : [0] encode_+ -> + [0] encode_g -> g [0] encode_a -> a [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_: -> null_encode_: [0] encode_+ -> null_encode_+ [0] encode_g -> null_encode_g [0] encode_a -> null_encode_a [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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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_g :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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_a :: map:c_app:cons:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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:filter2:true:filter:false:nil:::+:g:a:cons_app:null_encArg:null_encode_app:null_encode_::null_encode_+:null_encode_g:null_encode_a: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 => 8 cons => 3 filter2 => 6 true => 10 filter => 5 false => 4 nil => 9 : => 1 + => 0 g => 7 a => 2 null_encArg => 0 null_encode_app => 0 null_encode_: => 0 null_encode_+ => 0 null_encode_g => 0 null_encode_a => 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 }-> app(app(app(app(6, app(f, x)), f), x), xs) :|: f >= 0, xs >= 0, x >= 0, z'' = 1 + (1 + 3 + x) + xs, z' = 1 + 5 + f app(z', z'') -{ 1 }-> app(app(5, f), xs) :|: f >= 0, xs >= 0, z' = 1 + (1 + (1 + 6 + 4) + f) + x, x >= 0, z'' = xs app(z', z'') -{ 1 }-> app(app(3, x), app(app(5, f), xs)) :|: f >= 0, xs >= 0, z' = 1 + (1 + (1 + 6 + 10) + f) + x, x >= 0, z'' = xs app(z', z'') -{ 1 }-> app(app(3, app(f, x)), app(app(8, f), xs)) :|: f >= 0, xs >= 0, z' = 1 + 8 + f, x >= 0, z'' = 1 + (1 + 3 + x) + xs app(z', z'') -{ 1 }-> app(app(1, x), app(app(1, y), z)) :|: z'' = z, z >= 0, x >= 0, y >= 0, z' = 1 + 1 + (1 + (1 + 1 + x) + y) app(z', z'') -{ 1 }-> app(app(1, app(app(7, z), y)), app(app(0, x), 2)) :|: z' = 1 + 1 + z, f >= 0, z'' = 1 + (1 + 0 + x) + (1 + f + y), z >= 0, x >= 0, y >= 0 app(z', z'') -{ 1 }-> app(app(0, app(app(1, x), z)), app(app(1, y), z)) :|: z'' = z, z >= 0, z' = 1 + 1 + (1 + (1 + 0 + x) + y), x >= 0, y >= 0 app(z', z'') -{ 1 }-> 9 :|: f >= 0, z' = 1 + 8 + f, z'' = 9 app(z', z'') -{ 1 }-> 9 :|: f >= 0, z' = 1 + 5 + f, z'' = 9 app(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 app(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0 encArg(z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 9 :|: z' = 9 encArg(z') -{ 0 }-> 8 :|: z' = 8 encArg(z') -{ 0 }-> 7 :|: z' = 7 encArg(z') -{ 0 }-> 6 :|: z' = 6 encArg(z') -{ 0 }-> 5 :|: z' = 5 encArg(z') -{ 0 }-> 4 :|: z' = 4 encArg(z') -{ 0 }-> 3 :|: z' = 3 encArg(z') -{ 0 }-> 2 :|: z' = 2 encArg(z') -{ 0 }-> 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 :|: encode_: -{ 0 }-> 1 :|: encode_: -{ 0 }-> 0 :|: encode_a -{ 0 }-> 2 :|: encode_a -{ 0 }-> 0 :|: encode_app(z', z'') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_app(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_cons -{ 0 }-> 3 :|: encode_cons -{ 0 }-> 0 :|: encode_false -{ 0 }-> 4 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 5 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 6 :|: encode_filter2 -{ 0 }-> 0 :|: encode_g -{ 0 }-> 7 :|: encode_g -{ 0 }-> 0 :|: encode_map -{ 0 }-> 8 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 9 :|: encode_nil -{ 0 }-> 0 :|: encode_true -{ 0 }-> 10 :|: 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(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(:) -> c ENCARG(+) -> c1 ENCARG(g) -> c2 ENCARG(a) -> c3 ENCARG(map) -> c4 ENCARG(nil) -> c5 ENCARG(cons) -> c6 ENCARG(filter) -> c7 ENCARG(filter2) -> c8 ENCARG(true) -> c9 ENCARG(false) -> c10 ENCARG(cons_app(z0, z1)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_: -> c13 ENCODE_+ -> c14 ENCODE_G -> c15 ENCODE_A -> c16 ENCODE_MAP -> c17 ENCODE_NIL -> c18 ENCODE_CONS -> c19 ENCODE_FILTER -> c20 ENCODE_FILTER2 -> c21 ENCODE_TRUE -> c22 ENCODE_FALSE -> c23 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(:, z0), APP(app(:, z1), z2), APP(:, z1)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)), APP(+, app(app(:, z0), z2)), APP(app(:, z0), z2), APP(:, z0), APP(app(:, z1), z2), APP(:, z1)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26(APP(app(:, app(app(g, z0), z3)), app(app(+, z1), a)), APP(:, app(app(g, z0), z3)), APP(app(g, z0), z3), APP(g, z0), APP(app(+, z1), a), APP(+, z1)) APP(app(map, z0), nil) -> c27 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0)) APP(app(filter, z0), nil) -> c29 APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(:, z0), APP(app(:, z1), z2), APP(:, z1)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)), APP(+, app(app(:, z0), z2)), APP(app(:, z0), z2), APP(:, z0), APP(app(:, z1), z2), APP(:, z1)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26(APP(app(:, app(app(g, z0), z3)), app(app(+, z1), a)), APP(:, app(app(g, z0), z3)), APP(app(g, z0), z3), APP(g, z0), APP(app(+, z1), a), APP(+, z1)) APP(app(map, z0), nil) -> c27 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0)) APP(app(filter, z0), nil) -> c29 APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_:, encode_+, encode_g, encode_a, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, ENCODE_:, ENCODE_+, ENCODE_G, ENCODE_A, 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_3, c12_3, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24_4, c25_6, c26_6, c27, c28_5, c29, c30_5, c31_4, c32_2 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 22 trailing nodes: ENCODE_G -> c15 ENCODE_FALSE -> c23 ENCODE_NIL -> c18 ENCARG(filter) -> c7 ENCARG(map) -> c4 ENCODE_TRUE -> c22 ENCODE_CONS -> c19 ENCODE_: -> c13 ENCARG(false) -> c10 ENCARG(a) -> c3 ENCODE_MAP -> c17 ENCARG(true) -> c9 ENCARG(g) -> c2 ENCARG(:) -> c ENCODE_FILTER -> c20 ENCARG(cons) -> c6 ENCARG(nil) -> c5 ENCARG(+) -> c1 ENCODE_+ -> c14 ENCODE_A -> c16 ENCODE_FILTER2 -> c21 ENCARG(filter2) -> c8 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(:, z0), APP(app(:, z1), z2), APP(:, z1)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)), APP(+, app(app(:, z0), z2)), APP(app(:, z0), z2), APP(:, z0), APP(app(:, z1), z2), APP(:, z1)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26(APP(app(:, app(app(g, z0), z3)), app(app(+, z1), a)), APP(:, app(app(g, z0), z3)), APP(app(g, z0), z3), APP(g, z0), APP(app(+, z1), a), APP(+, z1)) APP(app(map, z0), nil) -> c27 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0)) APP(app(filter, z0), nil) -> c29 APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(:, z0), APP(app(:, z1), z2), APP(:, z1)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)), APP(+, app(app(:, z0), z2)), APP(app(:, z0), z2), APP(:, z0), APP(app(:, z1), z2), APP(:, z1)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26(APP(app(:, app(app(g, z0), z3)), app(app(+, z1), a)), APP(:, app(app(g, z0), z3)), APP(app(g, z0), z3), APP(g, z0), APP(app(+, z1), a), APP(+, z1)) APP(app(map, z0), nil) -> c27 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0)) APP(app(filter, z0), nil) -> c29 APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_:, encode_+, encode_g, encode_a, 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: c11_3, c12_3, c24_4, c25_6, c26_6, c27, c28_5, c29, c30_5, c31_4, c32_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 22 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_:, encode_+, encode_g, encode_a, 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: c11_3, c12_3, c27, c29, c24_2, c25_2, c26, c28_2, c30_2, c31_1, c32_1 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCODE_APP(z0, z1) -> c(ENCARG(z0)) ENCODE_APP(z0, z1) -> c(ENCARG(z1)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_:, encode_+, encode_g, encode_a, 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: c11_3, c27, c29, c24_2, c25_2, c26, c28_2, c30_2, c31_1, c32_1, c_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_APP(z0, z1) -> c(ENCARG(z0)) ENCODE_APP(z0, z1) -> c(ENCARG(z1)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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_: -> : encode_+ -> + encode_g -> g encode_a -> a encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_:, encode_+, encode_g, encode_a, 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: c11_3, c27, c29, c24_2, c25_2, c26, c28_2, c30_2, c31_1, c32_1, c_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_: -> : encode_+ -> + encode_g -> g encode_a -> a 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(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c11_3, c27, c29, c24_2, c25_2, c26, c28_2, c30_2, c31_1, c32_1, c_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0), ENCARG(:)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0), ENCARG(+)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0), ENCARG(g)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(:, x1)) -> c11(APP(:, encArg(x1)), ENCARG(:), ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(APP(+, encArg(x1)), ENCARG(+), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(APP(g, encArg(x1)), ENCARG(g), ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(APP(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0), ENCARG(:)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0), ENCARG(+)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0), ENCARG(g)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(:, x1)) -> c11(APP(:, encArg(x1)), ENCARG(:), ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(APP(+, encArg(x1)), ENCARG(+), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(APP(g, encArg(x1)), ENCARG(g), ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(APP(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c30_2, c31_1, c32_1, c_1, c11_3 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 33 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c30_2, c31_1, c32_1, c_1, c11_3, c11_2, c11_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, z0), app(app(cons, z1), z2)) -> c30(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) by APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2))), app(:, app(app(+, z0), z1))), z2), x2), APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(app(app(filter2, app(app(:, app(app(g, z0), z3)), app(app(+, z1), a))), app(:, z0)), app(app(+, z1), app(z2, z3))), x2), APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(map, z0)), nil), x2), APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2), APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(filter, z0)), nil), x2), APP(app(filter, z0), nil)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2), APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2))), app(:, app(app(+, z0), z1))), z2), x2), APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(app(app(filter2, app(app(:, app(app(g, z0), z3)), app(app(+, z1), a))), app(:, z0)), app(app(+, z1), app(z2, z3))), x2), APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(map, z0)), nil), x2), APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2), APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(filter, z0)), nil), x2), APP(app(filter, z0), nil)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2), APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2))), app(:, app(app(+, z0), z1))), z2), x2), APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(app(app(filter2, app(app(:, app(app(g, z0), z3)), app(app(+, z1), a))), app(:, z0)), app(app(+, z1), app(z2, z3))), x2), APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(map, z0)), nil), x2), APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2), APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(app(app(filter2, nil), app(filter, z0)), nil), x2), APP(app(filter, z0), nil)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2), APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c_1, c11_3, c11_2, c11_1, c30_2 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c_1, c11_3, c11_2, c11_1, c30_2, c30_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) by ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(:, x1) -> c(APP(:, encArg(x1))) ENCODE_APP(+, x1) -> c(APP(+, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(a, x1) -> c(APP(a, 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))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(:, x1) -> c(APP(:, encArg(x1))) ENCODE_APP(+, x1) -> c(APP(+, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(a, x1) -> c(APP(a, 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_3, c11_2, c11_1, c30_2, c30_1, c_1 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(a, x1) -> c(APP(a, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(:, x1) -> c(APP(:, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(+, x1) -> c(APP(+, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_3, c11_2, c11_1, c30_2, c30_1, c_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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(x1, x2))) -> c11(APP(:, app(encArg(x1), encArg(x2))), ENCARG(:), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(APP(+, app(encArg(x1), encArg(x2))), ENCARG(+), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(APP(g, app(encArg(x1), encArg(x2))), ENCARG(g), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(APP(a, app(encArg(x1), encArg(x2))), ENCARG(a), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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(x1, x2))) -> c11(APP(:, app(encArg(x1), encArg(x2))), ENCARG(:), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(APP(+, app(encArg(x1), encArg(x2))), ENCARG(+), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(APP(g, app(encArg(x1), encArg(x2))), ENCARG(g), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(APP(a, app(encArg(x1), encArg(x2))), ENCARG(a), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_3, c11_2, c11_1, c30_2, c30_1, c_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 22 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_3, c11_2, c11_1, c30_2, c30_1, c_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by ENCARG(cons_app(cons_app(x0, x1), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1)), ENCARG(:)) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1)), ENCARG(+)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1)), ENCARG(g)) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1)), ENCARG(a)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(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))) -> c11(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, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(APP(app(+, encArg(x1)), encArg(x2)), ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(APP(app(a, encArg(x1)), encArg(x2)), ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(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)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, x1), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1)), ENCARG(:)) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1)), ENCARG(+)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1)), ENCARG(g)) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1)), ENCARG(a)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(APP(app(+, encArg(x1)), encArg(x2)), ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(APP(app(a, encArg(x1)), encArg(x2)), ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(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)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 19 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, :)) -> c11(APP(encArg(x0), :), ENCARG(x0)) by ENCARG(cons_app(:, :)) -> c11(APP(:, :), ENCARG(:)) ENCARG(cons_app(+, :)) -> c11(APP(+, :), ENCARG(+)) ENCARG(cons_app(g, :)) -> c11(APP(g, :), ENCARG(g)) ENCARG(cons_app(a, :)) -> c11(APP(a, :), ENCARG(a)) ENCARG(cons_app(map, :)) -> c11(APP(map, :), ENCARG(map)) ENCARG(cons_app(nil, :)) -> c11(APP(nil, :), ENCARG(nil)) ENCARG(cons_app(cons, :)) -> c11(APP(cons, :), ENCARG(cons)) ENCARG(cons_app(filter, :)) -> c11(APP(filter, :), ENCARG(filter)) ENCARG(cons_app(filter2, :)) -> c11(APP(filter2, :), ENCARG(filter2)) ENCARG(cons_app(true, :)) -> c11(APP(true, :), ENCARG(true)) ENCARG(cons_app(false, :)) -> c11(APP(false, :), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), :)) -> c11(APP(app(encArg(z0), encArg(z1)), :), ENCARG(cons_app(z0, z1))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, :)) -> c11(APP(:, :), ENCARG(:)) ENCARG(cons_app(+, :)) -> c11(APP(+, :), ENCARG(+)) ENCARG(cons_app(g, :)) -> c11(APP(g, :), ENCARG(g)) ENCARG(cons_app(a, :)) -> c11(APP(a, :), ENCARG(a)) ENCARG(cons_app(map, :)) -> c11(APP(map, :), ENCARG(map)) ENCARG(cons_app(nil, :)) -> c11(APP(nil, :), ENCARG(nil)) ENCARG(cons_app(cons, :)) -> c11(APP(cons, :), ENCARG(cons)) ENCARG(cons_app(filter, :)) -> c11(APP(filter, :), ENCARG(filter)) ENCARG(cons_app(filter2, :)) -> c11(APP(filter2, :), ENCARG(filter2)) ENCARG(cons_app(true, :)) -> c11(APP(true, :), ENCARG(true)) ENCARG(cons_app(false, :)) -> c11(APP(false, :), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(false, :)) -> c11(APP(false, :), ENCARG(false)) ENCARG(cons_app(nil, :)) -> c11(APP(nil, :), ENCARG(nil)) ENCARG(cons_app(+, :)) -> c11(APP(+, :), ENCARG(+)) ENCARG(cons_app(true, :)) -> c11(APP(true, :), ENCARG(true)) ENCARG(cons_app(filter, :)) -> c11(APP(filter, :), ENCARG(filter)) ENCARG(cons_app(a, :)) -> c11(APP(a, :), ENCARG(a)) ENCARG(cons_app(:, :)) -> c11(APP(:, :), ENCARG(:)) ENCARG(cons_app(map, :)) -> c11(APP(map, :), ENCARG(map)) ENCARG(cons_app(cons, :)) -> c11(APP(cons, :), ENCARG(cons)) ENCARG(cons_app(g, :)) -> c11(APP(g, :), ENCARG(g)) ENCARG(cons_app(filter2, :)) -> c11(APP(filter2, :), ENCARG(filter2)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, +)) -> c11(APP(encArg(x0), +), ENCARG(x0)) by ENCARG(cons_app(:, +)) -> c11(APP(:, +), ENCARG(:)) ENCARG(cons_app(+, +)) -> c11(APP(+, +), ENCARG(+)) ENCARG(cons_app(g, +)) -> c11(APP(g, +), ENCARG(g)) ENCARG(cons_app(a, +)) -> c11(APP(a, +), ENCARG(a)) ENCARG(cons_app(map, +)) -> c11(APP(map, +), ENCARG(map)) ENCARG(cons_app(nil, +)) -> c11(APP(nil, +), ENCARG(nil)) ENCARG(cons_app(cons, +)) -> c11(APP(cons, +), ENCARG(cons)) ENCARG(cons_app(filter, +)) -> c11(APP(filter, +), ENCARG(filter)) ENCARG(cons_app(filter2, +)) -> c11(APP(filter2, +), ENCARG(filter2)) ENCARG(cons_app(true, +)) -> c11(APP(true, +), ENCARG(true)) ENCARG(cons_app(false, +)) -> c11(APP(false, +), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), +)) -> c11(APP(app(encArg(z0), encArg(z1)), +), ENCARG(cons_app(z0, z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, +)) -> c11(APP(:, +), ENCARG(:)) ENCARG(cons_app(+, +)) -> c11(APP(+, +), ENCARG(+)) ENCARG(cons_app(g, +)) -> c11(APP(g, +), ENCARG(g)) ENCARG(cons_app(a, +)) -> c11(APP(a, +), ENCARG(a)) ENCARG(cons_app(map, +)) -> c11(APP(map, +), ENCARG(map)) ENCARG(cons_app(nil, +)) -> c11(APP(nil, +), ENCARG(nil)) ENCARG(cons_app(cons, +)) -> c11(APP(cons, +), ENCARG(cons)) ENCARG(cons_app(filter, +)) -> c11(APP(filter, +), ENCARG(filter)) ENCARG(cons_app(filter2, +)) -> c11(APP(filter2, +), ENCARG(filter2)) ENCARG(cons_app(true, +)) -> c11(APP(true, +), ENCARG(true)) ENCARG(cons_app(false, +)) -> c11(APP(false, +), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(filter2, +)) -> c11(APP(filter2, +), ENCARG(filter2)) ENCARG(cons_app(+, +)) -> c11(APP(+, +), ENCARG(+)) ENCARG(cons_app(false, +)) -> c11(APP(false, +), ENCARG(false)) ENCARG(cons_app(cons, +)) -> c11(APP(cons, +), ENCARG(cons)) ENCARG(cons_app(map, +)) -> c11(APP(map, +), ENCARG(map)) ENCARG(cons_app(true, +)) -> c11(APP(true, +), ENCARG(true)) ENCARG(cons_app(filter, +)) -> c11(APP(filter, +), ENCARG(filter)) ENCARG(cons_app(a, +)) -> c11(APP(a, +), ENCARG(a)) ENCARG(cons_app(:, +)) -> c11(APP(:, +), ENCARG(:)) ENCARG(cons_app(nil, +)) -> c11(APP(nil, +), ENCARG(nil)) ENCARG(cons_app(g, +)) -> c11(APP(g, +), ENCARG(g)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, g)) -> c11(APP(encArg(x0), g), ENCARG(x0)) by ENCARG(cons_app(:, g)) -> c11(APP(:, g), ENCARG(:)) ENCARG(cons_app(+, g)) -> c11(APP(+, g), ENCARG(+)) ENCARG(cons_app(g, g)) -> c11(APP(g, g), ENCARG(g)) ENCARG(cons_app(a, g)) -> c11(APP(a, g), ENCARG(a)) ENCARG(cons_app(map, g)) -> c11(APP(map, g), ENCARG(map)) ENCARG(cons_app(nil, g)) -> c11(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(cons, g)) -> c11(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(filter, g)) -> c11(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(filter2, g)) -> c11(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(true, g)) -> c11(APP(true, g), ENCARG(true)) ENCARG(cons_app(false, g)) -> c11(APP(false, g), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), g)) -> c11(APP(app(encArg(z0), encArg(z1)), g), ENCARG(cons_app(z0, z1))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, g)) -> c11(APP(:, g), ENCARG(:)) ENCARG(cons_app(+, g)) -> c11(APP(+, g), ENCARG(+)) ENCARG(cons_app(g, g)) -> c11(APP(g, g), ENCARG(g)) ENCARG(cons_app(a, g)) -> c11(APP(a, g), ENCARG(a)) ENCARG(cons_app(map, g)) -> c11(APP(map, g), ENCARG(map)) ENCARG(cons_app(nil, g)) -> c11(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(cons, g)) -> c11(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(filter, g)) -> c11(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(filter2, g)) -> c11(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(true, g)) -> c11(APP(true, g), ENCARG(true)) ENCARG(cons_app(false, g)) -> c11(APP(false, g), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(map, g)) -> c11(APP(map, g), ENCARG(map)) ENCARG(cons_app(true, g)) -> c11(APP(true, g), ENCARG(true)) ENCARG(cons_app(:, g)) -> c11(APP(:, g), ENCARG(:)) ENCARG(cons_app(filter, g)) -> c11(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(nil, g)) -> c11(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(a, g)) -> c11(APP(a, g), ENCARG(a)) ENCARG(cons_app(cons, g)) -> c11(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(+, g)) -> c11(APP(+, g), ENCARG(+)) ENCARG(cons_app(false, g)) -> c11(APP(false, g), ENCARG(false)) ENCARG(cons_app(filter2, g)) -> c11(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(g, g)) -> c11(APP(g, g), ENCARG(g)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, a)) -> c11(APP(encArg(x0), a), ENCARG(x0)) by ENCARG(cons_app(:, a)) -> c11(APP(:, a), ENCARG(:)) ENCARG(cons_app(+, a)) -> c11(APP(+, a), ENCARG(+)) ENCARG(cons_app(g, a)) -> c11(APP(g, a), ENCARG(g)) ENCARG(cons_app(a, a)) -> c11(APP(a, a), ENCARG(a)) ENCARG(cons_app(map, a)) -> c11(APP(map, a), ENCARG(map)) ENCARG(cons_app(nil, a)) -> c11(APP(nil, a), ENCARG(nil)) ENCARG(cons_app(cons, a)) -> c11(APP(cons, a), ENCARG(cons)) ENCARG(cons_app(filter, a)) -> c11(APP(filter, a), ENCARG(filter)) ENCARG(cons_app(filter2, a)) -> c11(APP(filter2, a), ENCARG(filter2)) ENCARG(cons_app(true, a)) -> c11(APP(true, a), ENCARG(true)) ENCARG(cons_app(false, a)) -> c11(APP(false, a), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), a)) -> c11(APP(app(encArg(z0), encArg(z1)), a), ENCARG(cons_app(z0, z1))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, a)) -> c11(APP(:, a), ENCARG(:)) ENCARG(cons_app(+, a)) -> c11(APP(+, a), ENCARG(+)) ENCARG(cons_app(g, a)) -> c11(APP(g, a), ENCARG(g)) ENCARG(cons_app(a, a)) -> c11(APP(a, a), ENCARG(a)) ENCARG(cons_app(map, a)) -> c11(APP(map, a), ENCARG(map)) ENCARG(cons_app(nil, a)) -> c11(APP(nil, a), ENCARG(nil)) ENCARG(cons_app(cons, a)) -> c11(APP(cons, a), ENCARG(cons)) ENCARG(cons_app(filter, a)) -> c11(APP(filter, a), ENCARG(filter)) ENCARG(cons_app(filter2, a)) -> c11(APP(filter2, a), ENCARG(filter2)) ENCARG(cons_app(true, a)) -> c11(APP(true, a), ENCARG(true)) ENCARG(cons_app(false, a)) -> c11(APP(false, a), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(cons, a)) -> c11(APP(cons, a), ENCARG(cons)) ENCARG(cons_app(false, a)) -> c11(APP(false, a), ENCARG(false)) ENCARG(cons_app(filter2, a)) -> c11(APP(filter2, a), ENCARG(filter2)) ENCARG(cons_app(map, a)) -> c11(APP(map, a), ENCARG(map)) ENCARG(cons_app(g, a)) -> c11(APP(g, a), ENCARG(g)) ENCARG(cons_app(nil, a)) -> c11(APP(nil, a), ENCARG(nil)) ENCARG(cons_app(true, a)) -> c11(APP(true, a), ENCARG(true)) ENCARG(cons_app(+, a)) -> c11(APP(+, a), ENCARG(+)) ENCARG(cons_app(:, a)) -> c11(APP(:, a), ENCARG(:)) ENCARG(cons_app(filter, a)) -> c11(APP(filter, a), ENCARG(filter)) ENCARG(cons_app(a, a)) -> c11(APP(a, a), ENCARG(a)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) by ENCARG(cons_app(:, map)) -> c11(APP(:, map), ENCARG(:)) ENCARG(cons_app(+, map)) -> c11(APP(+, map), ENCARG(+)) ENCARG(cons_app(g, map)) -> c11(APP(g, map), ENCARG(g)) ENCARG(cons_app(a, map)) -> c11(APP(a, map), ENCARG(a)) ENCARG(cons_app(map, map)) -> c11(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c11(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c11(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c11(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c11(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c11(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c11(APP(false, map), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), map)) -> c11(APP(app(encArg(z0), encArg(z1)), map), ENCARG(cons_app(z0, z1))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, map)) -> c11(APP(:, map), ENCARG(:)) ENCARG(cons_app(+, map)) -> c11(APP(+, map), ENCARG(+)) ENCARG(cons_app(g, map)) -> c11(APP(g, map), ENCARG(g)) ENCARG(cons_app(a, map)) -> c11(APP(a, map), ENCARG(a)) ENCARG(cons_app(map, map)) -> c11(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c11(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c11(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c11(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c11(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c11(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c11(APP(false, map), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(true, map)) -> c11(APP(true, map), ENCARG(true)) ENCARG(cons_app(map, map)) -> c11(APP(map, map), ENCARG(map)) ENCARG(cons_app(false, map)) -> c11(APP(false, map), ENCARG(false)) ENCARG(cons_app(:, map)) -> c11(APP(:, map), ENCARG(:)) ENCARG(cons_app(+, map)) -> c11(APP(+, map), ENCARG(+)) ENCARG(cons_app(filter2, map)) -> c11(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(filter, map)) -> c11(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(a, map)) -> c11(APP(a, map), ENCARG(a)) ENCARG(cons_app(nil, map)) -> c11(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c11(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(g, map)) -> c11(APP(g, map), ENCARG(g)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) by ENCARG(cons_app(:, nil)) -> c11(APP(:, nil), ENCARG(:)) ENCARG(cons_app(+, nil)) -> c11(APP(+, nil), ENCARG(+)) ENCARG(cons_app(g, nil)) -> c11(APP(g, nil), ENCARG(g)) ENCARG(cons_app(a, nil)) -> c11(APP(a, nil), ENCARG(a)) ENCARG(cons_app(map, nil)) -> c11(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c11(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c11(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c11(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c11(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c11(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c11(APP(false, nil), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), nil)) -> c11(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, nil)) -> c11(APP(:, nil), ENCARG(:)) ENCARG(cons_app(+, nil)) -> c11(APP(+, nil), ENCARG(+)) ENCARG(cons_app(g, nil)) -> c11(APP(g, nil), ENCARG(g)) ENCARG(cons_app(a, nil)) -> c11(APP(a, nil), ENCARG(a)) ENCARG(cons_app(map, nil)) -> c11(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c11(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c11(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c11(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c11(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c11(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c11(APP(false, nil), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(cons, nil)) -> c11(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(:, nil)) -> c11(APP(:, nil), ENCARG(:)) ENCARG(cons_app(true, nil)) -> c11(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c11(APP(false, nil), ENCARG(false)) ENCARG(cons_app(map, nil)) -> c11(APP(map, nil), ENCARG(map)) ENCARG(cons_app(g, nil)) -> c11(APP(g, nil), ENCARG(g)) ENCARG(cons_app(filter, nil)) -> c11(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(a, nil)) -> c11(APP(a, nil), ENCARG(a)) ENCARG(cons_app(nil, nil)) -> c11(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(filter2, nil)) -> c11(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(+, nil)) -> c11(APP(+, nil), ENCARG(+)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) by ENCARG(cons_app(:, cons)) -> c11(APP(:, cons), ENCARG(:)) ENCARG(cons_app(+, cons)) -> c11(APP(+, cons), ENCARG(+)) ENCARG(cons_app(g, cons)) -> c11(APP(g, cons), ENCARG(g)) ENCARG(cons_app(a, cons)) -> c11(APP(a, cons), ENCARG(a)) ENCARG(cons_app(map, cons)) -> c11(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c11(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c11(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c11(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c11(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c11(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c11(APP(false, cons), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), cons)) -> c11(APP(app(encArg(z0), encArg(z1)), cons), ENCARG(cons_app(z0, z1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, cons)) -> c11(APP(:, cons), ENCARG(:)) ENCARG(cons_app(+, cons)) -> c11(APP(+, cons), ENCARG(+)) ENCARG(cons_app(g, cons)) -> c11(APP(g, cons), ENCARG(g)) ENCARG(cons_app(a, cons)) -> c11(APP(a, cons), ENCARG(a)) ENCARG(cons_app(map, cons)) -> c11(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c11(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c11(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c11(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c11(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c11(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c11(APP(false, cons), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(filter, cons)) -> c11(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(a, cons)) -> c11(APP(a, cons), ENCARG(a)) ENCARG(cons_app(false, cons)) -> c11(APP(false, cons), ENCARG(false)) ENCARG(cons_app(map, cons)) -> c11(APP(map, cons), ENCARG(map)) ENCARG(cons_app(:, cons)) -> c11(APP(:, cons), ENCARG(:)) ENCARG(cons_app(+, cons)) -> c11(APP(+, cons), ENCARG(+)) ENCARG(cons_app(cons, cons)) -> c11(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(true, cons)) -> c11(APP(true, cons), ENCARG(true)) ENCARG(cons_app(g, cons)) -> c11(APP(g, cons), ENCARG(g)) ENCARG(cons_app(filter2, cons)) -> c11(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(nil, cons)) -> c11(APP(nil, cons), ENCARG(nil)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) by ENCARG(cons_app(:, filter)) -> c11(APP(:, filter), ENCARG(:)) ENCARG(cons_app(+, filter)) -> c11(APP(+, filter), ENCARG(+)) ENCARG(cons_app(g, filter)) -> c11(APP(g, filter), ENCARG(g)) ENCARG(cons_app(a, filter)) -> c11(APP(a, filter), ENCARG(a)) ENCARG(cons_app(map, filter)) -> c11(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c11(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c11(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c11(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c11(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c11(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c11(APP(false, filter), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter)) -> c11(APP(app(encArg(z0), encArg(z1)), filter), ENCARG(cons_app(z0, z1))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, filter)) -> c11(APP(:, filter), ENCARG(:)) ENCARG(cons_app(+, filter)) -> c11(APP(+, filter), ENCARG(+)) ENCARG(cons_app(g, filter)) -> c11(APP(g, filter), ENCARG(g)) ENCARG(cons_app(a, filter)) -> c11(APP(a, filter), ENCARG(a)) ENCARG(cons_app(map, filter)) -> c11(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c11(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c11(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c11(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c11(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c11(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c11(APP(false, filter), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(cons, filter)) -> c11(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c11(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(a, filter)) -> c11(APP(a, filter), ENCARG(a)) ENCARG(cons_app(g, filter)) -> c11(APP(g, filter), ENCARG(g)) ENCARG(cons_app(nil, filter)) -> c11(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(+, filter)) -> c11(APP(+, filter), ENCARG(+)) ENCARG(cons_app(false, filter)) -> c11(APP(false, filter), ENCARG(false)) ENCARG(cons_app(map, filter)) -> c11(APP(map, filter), ENCARG(map)) ENCARG(cons_app(true, filter)) -> c11(APP(true, filter), ENCARG(true)) ENCARG(cons_app(:, filter)) -> c11(APP(:, filter), ENCARG(:)) ENCARG(cons_app(filter2, filter)) -> c11(APP(filter2, filter), ENCARG(filter2)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) by ENCARG(cons_app(:, filter2)) -> c11(APP(:, filter2), ENCARG(:)) ENCARG(cons_app(+, filter2)) -> c11(APP(+, filter2), ENCARG(+)) ENCARG(cons_app(g, filter2)) -> c11(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(a, filter2)) -> c11(APP(a, filter2), ENCARG(a)) ENCARG(cons_app(map, filter2)) -> c11(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c11(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c11(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c11(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c11(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c11(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c11(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter2)) -> c11(APP(app(encArg(z0), encArg(z1)), filter2), ENCARG(cons_app(z0, z1))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, filter2)) -> c11(APP(:, filter2), ENCARG(:)) ENCARG(cons_app(+, filter2)) -> c11(APP(+, filter2), ENCARG(+)) ENCARG(cons_app(g, filter2)) -> c11(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(a, filter2)) -> c11(APP(a, filter2), ENCARG(a)) ENCARG(cons_app(map, filter2)) -> c11(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c11(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c11(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c11(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c11(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c11(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c11(APP(false, filter2), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(map, filter2)) -> c11(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(false, filter2)) -> c11(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(:, filter2)) -> c11(APP(:, filter2), ENCARG(:)) ENCARG(cons_app(cons, filter2)) -> c11(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(true, filter2)) -> c11(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(+, filter2)) -> c11(APP(+, filter2), ENCARG(+)) ENCARG(cons_app(g, filter2)) -> c11(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(nil, filter2)) -> c11(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(filter2, filter2)) -> c11(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(filter, filter2)) -> c11(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(a, filter2)) -> c11(APP(a, filter2), ENCARG(a)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) by ENCARG(cons_app(:, true)) -> c11(APP(:, true), ENCARG(:)) ENCARG(cons_app(+, true)) -> c11(APP(+, true), ENCARG(+)) ENCARG(cons_app(g, true)) -> c11(APP(g, true), ENCARG(g)) ENCARG(cons_app(a, true)) -> c11(APP(a, true), ENCARG(a)) ENCARG(cons_app(map, true)) -> c11(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c11(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c11(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c11(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c11(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c11(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c11(APP(false, true), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), true)) -> c11(APP(app(encArg(z0), encArg(z1)), true), ENCARG(cons_app(z0, z1))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, true)) -> c11(APP(:, true), ENCARG(:)) ENCARG(cons_app(+, true)) -> c11(APP(+, true), ENCARG(+)) ENCARG(cons_app(g, true)) -> c11(APP(g, true), ENCARG(g)) ENCARG(cons_app(a, true)) -> c11(APP(a, true), ENCARG(a)) ENCARG(cons_app(map, true)) -> c11(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c11(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c11(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c11(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c11(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c11(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c11(APP(false, true), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(true, true)) -> c11(APP(true, true), ENCARG(true)) ENCARG(cons_app(filter, true)) -> c11(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(cons, true)) -> c11(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(a, true)) -> c11(APP(a, true), ENCARG(a)) ENCARG(cons_app(filter2, true)) -> c11(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(:, true)) -> c11(APP(:, true), ENCARG(:)) ENCARG(cons_app(map, true)) -> c11(APP(map, true), ENCARG(map)) ENCARG(cons_app(+, true)) -> c11(APP(+, true), ENCARG(+)) ENCARG(cons_app(false, true)) -> c11(APP(false, true), ENCARG(false)) ENCARG(cons_app(g, true)) -> c11(APP(g, true), ENCARG(g)) ENCARG(cons_app(nil, true)) -> c11(APP(nil, true), ENCARG(nil)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_2, c11_1, c30_2, c30_1, c_1, c11_3 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) by ENCARG(cons_app(:, false)) -> c11(APP(:, false), ENCARG(:)) ENCARG(cons_app(+, false)) -> c11(APP(+, false), ENCARG(+)) ENCARG(cons_app(g, false)) -> c11(APP(g, false), ENCARG(g)) ENCARG(cons_app(a, false)) -> c11(APP(a, false), ENCARG(a)) ENCARG(cons_app(map, false)) -> c11(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c11(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c11(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c11(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c11(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c11(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c11(APP(false, false), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), false)) -> c11(APP(app(encArg(z0), encArg(z1)), false), ENCARG(cons_app(z0, z1))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(:, false)) -> c11(APP(:, false), ENCARG(:)) ENCARG(cons_app(+, false)) -> c11(APP(+, false), ENCARG(+)) ENCARG(cons_app(g, false)) -> c11(APP(g, false), ENCARG(g)) ENCARG(cons_app(a, false)) -> c11(APP(a, false), ENCARG(a)) ENCARG(cons_app(map, false)) -> c11(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c11(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c11(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c11(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c11(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c11(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c11(APP(false, false), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_2, c30_1, c_1, c11_3, c11_2 ---------------------------------------- (109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(filter2, false)) -> c11(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(:, false)) -> c11(APP(:, false), ENCARG(:)) ENCARG(cons_app(nil, false)) -> c11(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(+, false)) -> c11(APP(+, false), ENCARG(+)) ENCARG(cons_app(false, false)) -> c11(APP(false, false), ENCARG(false)) ENCARG(cons_app(g, false)) -> c11(APP(g, false), ENCARG(g)) ENCARG(cons_app(true, false)) -> c11(APP(true, false), ENCARG(true)) ENCARG(cons_app(cons, false)) -> c11(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c11(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(map, false)) -> c11(APP(map, false), ENCARG(map)) ENCARG(cons_app(a, false)) -> c11(APP(a, false), ENCARG(a)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_2, c30_1, c_1, c11_3, c11_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2), APP(app(:, app(app(:, z0), z1)), z2)) by APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)))), app(:, app(app(:, x0), z0))), app(app(+, z1), app(z2, z3))), x3), APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)))), app(:, app(app(:, x0), z0))), app(app(+, z1), app(z2, z3))), x3), APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)))), app(:, app(app(:, x0), z0))), app(app(+, z1), app(z2, z3))), x3), APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, 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: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_2, c30_1, c_1, c11_3, c11_2 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_2, c30_1, c_1, c11_3, c11_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) by APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2))), app(:, app(app(+, z0), z1))), z2), x2)), app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, app(app(g, z0), z3)), app(app(+, z1), a))), app(:, z0)), app(app(+, z1), app(z2, z3))), x2)), app(filter, app(:, z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3), APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(map, z0)), nil), x2)), app(filter, app(map, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(map, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(filter, z0)), nil), x2)), app(filter, app(filter, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2))), app(:, app(app(+, z0), z1))), z2), x2)), app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, app(app(g, z0), z3)), app(app(+, z1), a))), app(:, z0)), app(app(+, z1), app(z2, z3))), x2)), app(filter, app(:, z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3), APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(map, z0)), nil), x2)), app(filter, app(map, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(map, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(filter, z0)), nil), x2)), app(filter, app(filter, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2))), app(:, app(app(+, z0), z1))), z2), x2)), app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, app(app(g, z0), z3)), app(app(+, z1), a))), app(:, z0)), app(app(+, z1), app(z2, z3))), x2)), app(filter, app(:, z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3), APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(map, z0)), nil), x2)), app(filter, app(map, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(map, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, nil), app(filter, z0)), nil), x2)), app(filter, app(filter, z0))), app(app(cons, nil), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_2, c30_1, c_1, c11_3, c11_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_2, c30_1, c_1, c11_3, c11_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) by APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, nil), app(app(app(filter2, false), z0), x1)), nil), x3), APP(app(app(app(filter2, false), z0), x1), nil)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, nil), app(app(app(filter2, false), z0), x1)), nil), x3), APP(app(app(app(filter2, false), z0), x1), nil)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, nil), app(app(app(filter2, false), z0), x1)), nil), x3), APP(app(app(app(filter2, false), z0), x1), nil)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, :) -> c(APP(encArg(x0), :)) by ENCODE_APP(:, :) -> c(APP(:, :)) ENCODE_APP(+, :) -> c(APP(+, :)) ENCODE_APP(g, :) -> c(APP(g, :)) ENCODE_APP(a, :) -> c(APP(a, :)) ENCODE_APP(map, :) -> c(APP(map, :)) ENCODE_APP(nil, :) -> c(APP(nil, :)) ENCODE_APP(cons, :) -> c(APP(cons, :)) ENCODE_APP(filter, :) -> c(APP(filter, :)) ENCODE_APP(filter2, :) -> c(APP(filter2, :)) ENCODE_APP(true, :) -> c(APP(true, :)) ENCODE_APP(false, :) -> c(APP(false, :)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(:, :) -> c(APP(:, :)) ENCODE_APP(+, :) -> c(APP(+, :)) ENCODE_APP(g, :) -> c(APP(g, :)) ENCODE_APP(a, :) -> c(APP(a, :)) ENCODE_APP(map, :) -> c(APP(map, :)) ENCODE_APP(nil, :) -> c(APP(nil, :)) ENCODE_APP(cons, :) -> c(APP(cons, :)) ENCODE_APP(filter, :) -> c(APP(filter, :)) ENCODE_APP(filter2, :) -> c(APP(filter2, :)) ENCODE_APP(true, :) -> c(APP(true, :)) ENCODE_APP(false, :) -> c(APP(false, :)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCODE_APP(true, :) -> c(APP(true, :)) ENCODE_APP(filter2, :) -> c(APP(filter2, :)) ENCODE_APP(nil, :) -> c(APP(nil, :)) ENCODE_APP(g, :) -> c(APP(g, :)) ENCODE_APP(map, :) -> c(APP(map, :)) ENCODE_APP(a, :) -> c(APP(a, :)) ENCODE_APP(false, :) -> c(APP(false, :)) ENCODE_APP(+, :) -> c(APP(+, :)) ENCODE_APP(:, :) -> c(APP(:, :)) ENCODE_APP(cons, :) -> c(APP(cons, :)) ENCODE_APP(filter, :) -> c(APP(filter, :)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, +) -> c(APP(encArg(x0), +)) by ENCODE_APP(:, +) -> c(APP(:, +)) ENCODE_APP(+, +) -> c(APP(+, +)) ENCODE_APP(g, +) -> c(APP(g, +)) ENCODE_APP(a, +) -> c(APP(a, +)) ENCODE_APP(map, +) -> c(APP(map, +)) ENCODE_APP(nil, +) -> c(APP(nil, +)) ENCODE_APP(cons, +) -> c(APP(cons, +)) ENCODE_APP(filter, +) -> c(APP(filter, +)) ENCODE_APP(filter2, +) -> c(APP(filter2, +)) ENCODE_APP(true, +) -> c(APP(true, +)) ENCODE_APP(false, +) -> c(APP(false, +)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(:, +) -> c(APP(:, +)) ENCODE_APP(+, +) -> c(APP(+, +)) ENCODE_APP(g, +) -> c(APP(g, +)) ENCODE_APP(a, +) -> c(APP(a, +)) ENCODE_APP(map, +) -> c(APP(map, +)) ENCODE_APP(nil, +) -> c(APP(nil, +)) ENCODE_APP(cons, +) -> c(APP(cons, +)) ENCODE_APP(filter, +) -> c(APP(filter, +)) ENCODE_APP(filter2, +) -> c(APP(filter2, +)) ENCODE_APP(true, +) -> c(APP(true, +)) ENCODE_APP(false, +) -> c(APP(false, +)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCODE_APP(cons, +) -> c(APP(cons, +)) ENCODE_APP(:, +) -> c(APP(:, +)) ENCODE_APP(filter2, +) -> c(APP(filter2, +)) ENCODE_APP(map, +) -> c(APP(map, +)) ENCODE_APP(false, +) -> c(APP(false, +)) ENCODE_APP(filter, +) -> c(APP(filter, +)) ENCODE_APP(a, +) -> c(APP(a, +)) ENCODE_APP(g, +) -> c(APP(g, +)) ENCODE_APP(nil, +) -> c(APP(nil, +)) ENCODE_APP(true, +) -> c(APP(true, +)) ENCODE_APP(+, +) -> c(APP(+, +)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) by ENCODE_APP(:, g) -> c(APP(:, g)) ENCODE_APP(+, g) -> c(APP(+, g)) ENCODE_APP(g, g) -> c(APP(g, g)) ENCODE_APP(a, g) -> c(APP(a, g)) ENCODE_APP(map, g) -> c(APP(map, g)) ENCODE_APP(nil, g) -> c(APP(nil, g)) ENCODE_APP(cons, g) -> c(APP(cons, g)) ENCODE_APP(filter, g) -> c(APP(filter, g)) ENCODE_APP(filter2, g) -> c(APP(filter2, g)) ENCODE_APP(true, g) -> c(APP(true, g)) ENCODE_APP(false, g) -> c(APP(false, g)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) ENCODE_APP(:, g) -> c(APP(:, g)) ENCODE_APP(+, g) -> c(APP(+, g)) ENCODE_APP(g, g) -> c(APP(g, g)) ENCODE_APP(a, g) -> c(APP(a, g)) ENCODE_APP(map, g) -> c(APP(map, g)) ENCODE_APP(nil, g) -> c(APP(nil, g)) ENCODE_APP(cons, g) -> c(APP(cons, g)) ENCODE_APP(filter, g) -> c(APP(filter, g)) ENCODE_APP(filter2, g) -> c(APP(filter2, g)) ENCODE_APP(true, g) -> c(APP(true, g)) ENCODE_APP(false, g) -> c(APP(false, g)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCODE_APP(map, g) -> c(APP(map, g)) ENCODE_APP(nil, g) -> c(APP(nil, g)) ENCODE_APP(false, g) -> c(APP(false, g)) ENCODE_APP(g, g) -> c(APP(g, g)) ENCODE_APP(cons, g) -> c(APP(cons, g)) ENCODE_APP(true, g) -> c(APP(true, g)) ENCODE_APP(filter2, g) -> c(APP(filter2, g)) ENCODE_APP(+, g) -> c(APP(+, g)) ENCODE_APP(a, g) -> c(APP(a, g)) ENCODE_APP(:, g) -> c(APP(:, g)) ENCODE_APP(filter, g) -> c(APP(filter, g)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) 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(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, a) -> c(APP(encArg(x0), a)) by ENCODE_APP(:, a) -> c(APP(:, a)) ENCODE_APP(+, a) -> c(APP(+, a)) ENCODE_APP(g, a) -> c(APP(g, a)) ENCODE_APP(a, a) -> c(APP(a, a)) ENCODE_APP(map, a) -> c(APP(map, a)) ENCODE_APP(nil, a) -> c(APP(nil, a)) ENCODE_APP(cons, a) -> c(APP(cons, a)) ENCODE_APP(filter, a) -> c(APP(filter, a)) ENCODE_APP(filter2, a) -> c(APP(filter2, a)) ENCODE_APP(true, a) -> c(APP(true, a)) ENCODE_APP(false, a) -> c(APP(false, a)) ENCODE_APP(cons_app(z0, z1), a) -> c(APP(app(encArg(z0), encArg(z1)), a)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(:, a) -> c(APP(:, a)) ENCODE_APP(+, a) -> c(APP(+, a)) ENCODE_APP(g, a) -> c(APP(g, a)) ENCODE_APP(a, a) -> c(APP(a, a)) ENCODE_APP(map, a) -> c(APP(map, a)) ENCODE_APP(nil, a) -> c(APP(nil, a)) ENCODE_APP(cons, a) -> c(APP(cons, a)) ENCODE_APP(filter, a) -> c(APP(filter, a)) ENCODE_APP(filter2, a) -> c(APP(filter2, a)) ENCODE_APP(true, a) -> c(APP(true, a)) ENCODE_APP(false, a) -> c(APP(false, a)) ENCODE_APP(cons_app(z0, z1), a) -> c(APP(app(encArg(z0), encArg(z1)), a)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCODE_APP(map, a) -> c(APP(map, a)) ENCODE_APP(g, a) -> c(APP(g, a)) ENCODE_APP(false, a) -> c(APP(false, a)) ENCODE_APP(+, a) -> c(APP(+, a)) ENCODE_APP(cons, a) -> c(APP(cons, a)) ENCODE_APP(:, a) -> c(APP(:, a)) ENCODE_APP(filter, a) -> c(APP(filter, a)) ENCODE_APP(a, a) -> c(APP(a, a)) ENCODE_APP(true, a) -> c(APP(true, a)) ENCODE_APP(filter2, a) -> c(APP(filter2, a)) ENCODE_APP(nil, a) -> c(APP(nil, a)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), a) -> c(APP(app(encArg(z0), encArg(z1)), a)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) by ENCODE_APP(:, map) -> c(APP(:, map)) ENCODE_APP(+, map) -> c(APP(+, map)) ENCODE_APP(g, map) -> c(APP(g, map)) ENCODE_APP(a, map) -> c(APP(a, map)) ENCODE_APP(map, map) -> c(APP(map, map)) ENCODE_APP(nil, map) -> c(APP(nil, map)) ENCODE_APP(cons, map) -> c(APP(cons, map)) ENCODE_APP(filter, map) -> c(APP(filter, map)) ENCODE_APP(filter2, map) -> c(APP(filter2, map)) ENCODE_APP(true, map) -> c(APP(true, map)) ENCODE_APP(false, map) -> c(APP(false, map)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), a) -> c(APP(app(encArg(z0), encArg(z1)), a)) ENCODE_APP(:, map) -> c(APP(:, map)) ENCODE_APP(+, map) -> c(APP(+, map)) ENCODE_APP(g, map) -> c(APP(g, map)) ENCODE_APP(a, map) -> c(APP(a, 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCODE_APP(:, map) -> c(APP(:, map)) ENCODE_APP(a, map) -> c(APP(a, map)) ENCODE_APP(true, map) -> c(APP(true, map)) ENCODE_APP(cons, map) -> c(APP(cons, map)) ENCODE_APP(nil, map) -> c(APP(nil, map)) ENCODE_APP(map, map) -> c(APP(map, map)) ENCODE_APP(g, map) -> c(APP(g, map)) ENCODE_APP(+, map) -> c(APP(+, map)) ENCODE_APP(false, map) -> c(APP(false, map)) ENCODE_APP(filter, map) -> c(APP(filter, map)) ENCODE_APP(filter2, map) -> c(APP(filter2, map)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), a) -> c(APP(app(encArg(z0), encArg(z1)), a)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) by ENCODE_APP(:, nil) -> c(APP(:, nil)) ENCODE_APP(+, nil) -> c(APP(+, nil)) ENCODE_APP(g, nil) -> c(APP(g, nil)) ENCODE_APP(a, nil) -> c(APP(a, nil)) ENCODE_APP(map, nil) -> c(APP(map, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(filter2, nil) -> c(APP(filter2, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), a) -> c(APP(app(encArg(z0), encArg(z1)), a)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(:, nil) -> c(APP(:, nil)) ENCODE_APP(+, nil) -> c(APP(+, nil)) ENCODE_APP(g, nil) -> c(APP(g, nil)) ENCODE_APP(a, nil) -> c(APP(a, nil)) ENCODE_APP(map, nil) -> c(APP(map, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(filter2, nil) -> c(APP(filter2, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2 ---------------------------------------- (145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCODE_APP(a, nil) -> c(APP(a, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(map, nil) -> c(APP(map, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(g, nil) -> c(APP(g, nil)) ENCODE_APP(+, nil) -> c(APP(+, nil)) ENCODE_APP(:, nil) -> c(APP(:, nil)) ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(filter2, nil) -> c(APP(filter2, nil)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: encArg(:) -> : encArg(+) -> + encArg(g) -> g encArg(a) -> a 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(app(:, app(app(:, z0), z1)), z2) -> app(app(:, z0), app(app(:, z1), z2)) app(app(:, app(app(+, z0), z1)), z2) -> app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)) app(app(:, z0), app(app(+, z1), app(z2, z3))) -> app(app(:, app(app(g, z0), z3)), app(app(+, z1), a)) 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(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) ENCARG(cons_app(:, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(+, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(a, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, :))) -> c11(APP(encArg(x0), app(encArg(x1), :)), ENCARG(x0), ENCARG(cons_app(x1, :))) ENCARG(cons_app(x0, cons_app(x1, +))) -> c11(APP(encArg(x0), app(encArg(x1), +)), ENCARG(x0), ENCARG(cons_app(x1, +))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c11(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, a))) -> c11(APP(encArg(x0), app(encArg(x1), a)), ENCARG(x0), ENCARG(cons_app(x1, a))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(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)))) -> c11(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(:, x2))) -> c11(APP(encArg(x0), app(:, encArg(x2))), ENCARG(x0), ENCARG(cons_app(:, x2))) ENCARG(cons_app(x0, cons_app(+, x2))) -> c11(APP(encArg(x0), app(+, encArg(x2))), ENCARG(x0), ENCARG(cons_app(+, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c11(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(a, x2))) -> c11(APP(encArg(x0), app(a, encArg(x2))), ENCARG(x0), ENCARG(cons_app(a, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(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))) -> c11(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))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(:, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(+, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(a, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, :), x2)) -> c11(APP(app(encArg(x0), :), encArg(x2)), ENCARG(cons_app(x0, :)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, +), x2)) -> c11(APP(app(encArg(x0), +), encArg(x2)), ENCARG(cons_app(x0, +)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c11(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, a), x2)) -> c11(APP(app(encArg(x0), a), encArg(x2)), ENCARG(cons_app(x0, a)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(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)) -> c11(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(:, x1), x2)) -> c11(APP(app(:, encArg(x1)), encArg(x2)), ENCARG(cons_app(:, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(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)) -> c11(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), :)) -> c11(APP(app(encArg(x0), encArg(x1)), :), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), +)) -> c11(APP(app(encArg(x0), encArg(x1)), +), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c11(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), a)) -> c11(APP(app(encArg(x0), encArg(x1)), a), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(+, x1), x2)) -> c11(ENCARG(cons_app(+, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c11(ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(a, x1), x2)) -> c11(ENCARG(cons_app(a, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), :) -> c(APP(app(encArg(z0), encArg(z1)), :)) ENCODE_APP(cons_app(z0, z1), +) -> c(APP(app(encArg(z0), encArg(z1)), +)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), a) -> c(APP(app(encArg(z0), encArg(z1)), a)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) S tuples: APP(app(map, z0), nil) -> c27 APP(app(filter, z0), nil) -> c29 APP(app(:, app(app(:, z0), z1)), z2) -> c24(APP(app(:, z0), app(app(:, z1), z2)), APP(app(:, z1), z2)) APP(app(:, app(app(+, z0), z1)), z2) -> c25(APP(app(:, z0), z2), APP(app(:, z1), z2)) APP(app(:, z0), app(app(+, z1), app(z2, z3))) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c28(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c32(APP(app(filter, z0), z2)) APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2)) -> c30(APP(app(:, app(app(+, z0), z1)), z2)) APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2)) -> c30(APP(app(:, z0), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c30(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c30(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c30(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c30(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(:, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(:, z0), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(:, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(:, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), app(app(+, z0), z1)))), app(app(cons, z2), x3)) -> c30(APP(app(app(app(filter2, app(app(:, x0), app(app(+, app(app(:, z0), z2)), app(app(:, z1), z2)))), app(:, app(app(:, x0), app(app(+, z0), z1)))), z2), x3), APP(app(:, app(app(:, x0), app(app(+, z0), z1))), z2)) APP(app(filter, app(:, app(app(:, x0), x1))), app(app(cons, x2), x3)) -> c30(APP(app(:, app(app(:, x0), x1)), x2)) APP(app(filter, app(:, app(app(:, x0), z0))), app(app(cons, app(app(+, z1), app(z2, z3))), x3)) -> c30(APP(app(:, app(app(:, x0), z0)), app(app(+, z1), app(z2, z3)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2)), app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, z2), x2)), x3), APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(:, z0), app(app(:, z1), z2))), app(:, app(app(:, z0), z1))), z2), x2)), app(filter, app(:, app(app(:, z0), z1)))), app(app(cons, z2), x2)), x3), APP(app(filter, app(:, app(app(:, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2)), app(filter, app(filter, z0))), app(app(cons, app(app(cons, z1), z2)), x2)), x3), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, app(app(+, z0), z1)))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c30(APP(app(filter, app(:, app(app(+, z0), z1))), app(app(cons, z2), x2))) APP(app(filter, app(filter, app(:, z0))), app(app(cons, app(app(cons, app(app(+, z1), app(z2, z3))), x2)), x3)) -> c30(APP(app(filter, app(:, z0)), app(app(cons, app(app(+, z1), app(z2, z3))), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c30(APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c30(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, app(app(cons, z1), z2)), x3)) -> c30(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(app(app(filter2, false), z0), x1)), app(app(cons, z1), z2)), x3), APP(app(app(app(filter2, false), z0), x1), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), x0), x1)), app(app(cons, x2), x3)) -> c30(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c30(APP(app(app(app(filter2, false), z0), x1), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c27, c29, c24_2, c25_2, c26, c28_2, c31_1, c32_1, c11_1, c30_1, c_1, c11_3, c11_2, c30_2