/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 183 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), 8838 ms] (14) BOUNDS(1, INF) (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (18) CdtProblem (19) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CdtProblem (21) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CdtProblem (23) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (26) CdtProblem (27) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 7 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 28 ms] (42) CdtProblem (43) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 56 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 40 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (126) CdtProblem (127) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (130) CpxRelTRS (131) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CpxWeightedTrs (133) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CpxTypedWeightedTrs (135) CompletionProof [UPPER BOUND(ID), 0 ms] (136) CpxTypedWeightedCompleteTrs (137) NarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (138) CpxTypedWeightedCompleteTrs (139) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms] (140) CpxRNTS (141) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CpxRNTS (143) CompletionProof [UPPER BOUND(ID), 0 ms] (144) CpxTypedWeightedCompleteTrs (145) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms] (146) CpxRNTS ---------------------------------------- (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(f, app(f, x)) -> app(g, app(f, x)) app(g, app(g, x)) -> app(f, x) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), 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(f) -> f encArg(g) -> g 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_f -> f encode_g -> g 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(f, app(f, x)) -> app(g, app(f, x)) app(g, app(g, x)) -> app(f, x) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g 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(f, app(f, x)) -> app(g, app(f, x)) app(g, app(g, x)) -> app(f, x) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g 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(f, app(f, x)) -> app(g, app(f, x)) app(g, app(g, x)) -> app(f, x) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g 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(f, app(f, x)) -> app(g, app(f, x)) app(g, app(g, x)) -> app(f, x) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g 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(f, app(f, x)) -> app(g, app(f, x)) app(g, app(g, x)) -> app(f, x) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app f :: f:g:map:nil:cons:filter:filter2:true:false:cons_app g :: f:g:map:nil:cons:filter:filter2:true:false:cons_app map :: f:g:map:nil:cons:filter:filter2:true:false:cons_app nil :: f:g:map:nil:cons:filter:filter2:true:false:cons_app cons :: f:g:map:nil:cons:filter:filter2:true:false:cons_app filter :: f:g:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: f:g:map:nil:cons:filter:filter2:true:false:cons_app true :: f:g:map:nil:cons:filter:filter2:true:false:cons_app false :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encArg :: f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_f :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_g :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: f:g:map:nil:cons:filter:filter2:true:false:cons_app hole_f:g:map:nil:cons:filter:filter2:true:false:cons_app1_0 :: f:g:map:nil:cons:filter:filter2:true:false:cons_app gen_f:g:map:nil:cons:filter:filter2:true:false:cons_app2_0 :: Nat -> f:g: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(f, app(f, x)) -> app(g, app(f, x)) app(g, app(g, x)) -> app(f, x) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app f :: f:g:map:nil:cons:filter:filter2:true:false:cons_app g :: f:g:map:nil:cons:filter:filter2:true:false:cons_app map :: f:g:map:nil:cons:filter:filter2:true:false:cons_app nil :: f:g:map:nil:cons:filter:filter2:true:false:cons_app cons :: f:g:map:nil:cons:filter:filter2:true:false:cons_app filter :: f:g:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: f:g:map:nil:cons:filter:filter2:true:false:cons_app true :: f:g:map:nil:cons:filter:filter2:true:false:cons_app false :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encArg :: f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app -> f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_f :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_g :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: f:g:map:nil:cons:filter:filter2:true:false:cons_app hole_f:g:map:nil:cons:filter:filter2:true:false:cons_app1_0 :: f:g:map:nil:cons:filter:filter2:true:false:cons_app gen_f:g:map:nil:cons:filter:filter2:true:false:cons_app2_0 :: Nat -> f:g:map:nil:cons:filter:filter2:true:false:cons_app Generator Equations: gen_f:g:map:nil:cons:filter:filter2:true:false:cons_app2_0(0) <=> f gen_f:g:map:nil:cons:filter:filter2:true:false:cons_app2_0(+(x, 1)) <=> cons_app(f, gen_f:g: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_f:g:map:nil:cons:filter:filter2:true:false:cons_app2_0(n2264_0)) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_f:g:map:nil:cons:filter:filter2:true:false:cons_app2_0(0)) Induction Step: encArg(gen_f:g:map:nil:cons:filter:filter2:true:false:cons_app2_0(+(n2264_0, 1))) ->_R^Omega(0) app(encArg(f), encArg(gen_f:g:map:nil:cons:filter:filter2:true:false:cons_app2_0(n2264_0))) ->_R^Omega(0) app(f, encArg(gen_f:g:map:nil:cons:filter:filter2:true:false:cons_app2_0(n2264_0))) ->_IH app(f, *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(f, app(f, x)) -> app(g, app(f, x)) app(g, app(g, x)) -> app(f, x) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g 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) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(f) -> c ENCARG(g) -> c1 ENCARG(map) -> c2 ENCARG(nil) -> c3 ENCARG(cons) -> c4 ENCARG(filter) -> c5 ENCARG(filter2) -> c6 ENCARG(true) -> c7 ENCARG(false) -> c8 ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F -> c11 ENCODE_G -> c12 ENCODE_MAP -> c13 ENCODE_NIL -> c14 ENCODE_CONS -> c15 ENCODE_FILTER -> c16 ENCODE_FILTER2 -> c17 ENCODE_TRUE -> c18 ENCODE_FALSE -> c19 APP(f, app(f, z0)) -> c20(APP(g, app(f, z0)), APP(f, z0)) APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(map, z0), app(app(cons, z1), z2)) -> c23(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) -> c24 APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(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) -> c26(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) -> c27(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(f, app(f, z0)) -> c20(APP(g, app(f, z0)), APP(f, z0)) APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(map, z0), app(app(cons, z1), z2)) -> c23(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) -> c24 APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(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) -> c26(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) -> c27(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_g, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, ENCODE_F, ENCODE_G, 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_3, c10_3, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20_2, c21_1, c22, c23_5, c24, c25_5, c26_4, c27_2 ---------------------------------------- (19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 18 trailing nodes: ENCODE_TRUE -> c18 ENCARG(true) -> c7 ENCARG(map) -> c2 ENCODE_G -> c12 ENCODE_CONS -> c15 ENCARG(nil) -> c3 ENCODE_FILTER2 -> c17 ENCARG(f) -> c ENCODE_NIL -> c14 ENCODE_FALSE -> c19 ENCODE_F -> c11 ENCODE_FILTER -> c16 ENCODE_MAP -> c13 ENCARG(filter2) -> c6 ENCARG(filter) -> c5 ENCARG(false) -> c8 ENCARG(g) -> c1 ENCARG(cons) -> c4 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(f, app(f, z0)) -> c20(APP(g, app(f, z0)), APP(f, z0)) APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(map, z0), app(app(cons, z1), z2)) -> c23(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) -> c24 APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(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) -> c26(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) -> c27(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(f, app(f, z0)) -> c20(APP(g, app(f, z0)), APP(f, z0)) APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(map, z0), app(app(cons, z1), z2)) -> c23(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) -> c24 APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(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) -> c26(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) -> c27(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_g, 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: c9_3, c10_3, c20_2, c21_1, c22, c23_5, c24, c25_5, c26_4, c27_2 ---------------------------------------- (21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing tuple parts ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_g, 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: c9_3, c10_3, c21_1, c22, c24, c20_1, c23_2, c25_2, c26_1, c27_1 ---------------------------------------- (23) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_g, 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: c9_3, c21_1, c22, c24, c20_1, c23_2, c25_2, c26_1, c27_1, c_1 ---------------------------------------- (25) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_APP(z0, z1) -> c(ENCARG(z0)) ENCODE_APP(z0, z1) -> c(ENCARG(z1)) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_g, 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: c9_3, c21_1, c22, c24, c20_1, c23_2, c25_2, c26_1, c27_1, c_1 ---------------------------------------- (27) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(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: c9_3, c21_1, c22, c24, c20_1, c23_2, c25_2, c26_1, c27_1, c_1 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0), ENCARG(f)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0), ENCARG(g)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(f), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(g), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0), ENCARG(f)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0), ENCARG(g)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(f), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(g), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(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: c21_1, c22, c24, c20_1, c23_2, c25_2, c26_1, c27_1, c_1, c9_3 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 25 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(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: c21_1, c22, c24, c20_1, c23_2, c25_2, c26_1, c27_1, c_1, c9_3, c9_2, c9_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, z0), app(app(cons, z1), z2)) -> c25(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) by APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(app(app(app(filter2, app(g, app(f, z0))), f), app(f, z0)), x2), APP(f, app(f, z0))) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(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)) -> c25(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)) -> c25(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)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(app(app(app(filter2, app(g, app(f, z0))), f), app(f, z0)), x2), APP(f, app(f, z0))) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(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)) -> c25(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)) -> c25(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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(app(app(app(filter2, app(g, app(f, z0))), f), app(f, z0)), x2), APP(f, app(f, z0))) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(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)) -> c25(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)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c_1, c9_3, c9_2, c9_1, c25_2 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c_1, c9_3, c9_2, c9_1, c25_2, c25_1 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) by ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, 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))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_3, c9_2, c9_1, c25_2, c25_1, c_1 ---------------------------------------- (39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_3, c9_2, c9_1, c25_2, c25_1, c_1 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(f), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(g), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(f), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(g), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_3, c9_2, c9_1, c25_2, c25_1, c_1 ---------------------------------------- (43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing tuple parts ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_3, c9_2, c9_1, c25_2, c25_1, c_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by ENCARG(cons_app(cons_app(x0, x1), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1)), ENCARG(f)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1)), ENCARG(g)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(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))) -> c9(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, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(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)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1)), ENCARG(f)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1)), ENCARG(g)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(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)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 14 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, f)) -> c9(APP(encArg(x0), f), ENCARG(x0)) by ENCARG(cons_app(f, f)) -> c9(APP(f, f), ENCARG(f)) ENCARG(cons_app(g, f)) -> c9(APP(g, f), ENCARG(g)) ENCARG(cons_app(map, f)) -> c9(APP(map, f), ENCARG(map)) ENCARG(cons_app(nil, f)) -> c9(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(cons, f)) -> c9(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(filter, f)) -> c9(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(filter2, f)) -> c9(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(true, f)) -> c9(APP(true, f), ENCARG(true)) ENCARG(cons_app(false, f)) -> c9(APP(false, f), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), f)) -> c9(APP(app(encArg(z0), encArg(z1)), f), ENCARG(cons_app(z0, z1))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, f)) -> c9(APP(f, f), ENCARG(f)) ENCARG(cons_app(g, f)) -> c9(APP(g, f), ENCARG(g)) ENCARG(cons_app(map, f)) -> c9(APP(map, f), ENCARG(map)) ENCARG(cons_app(nil, f)) -> c9(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(cons, f)) -> c9(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(filter, f)) -> c9(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(filter2, f)) -> c9(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(true, f)) -> c9(APP(true, f), ENCARG(true)) ENCARG(cons_app(false, f)) -> c9(APP(false, f), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(cons, f)) -> c9(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(g, f)) -> c9(APP(g, f), ENCARG(g)) ENCARG(cons_app(f, f)) -> c9(APP(f, f), ENCARG(f)) ENCARG(cons_app(nil, f)) -> c9(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(map, f)) -> c9(APP(map, f), ENCARG(map)) ENCARG(cons_app(filter2, f)) -> c9(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(filter, f)) -> c9(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(false, f)) -> c9(APP(false, f), ENCARG(false)) ENCARG(cons_app(true, f)) -> c9(APP(true, f), ENCARG(true)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, g)) -> c9(APP(encArg(x0), g), ENCARG(x0)) by ENCARG(cons_app(f, g)) -> c9(APP(f, g), ENCARG(f)) ENCARG(cons_app(g, g)) -> c9(APP(g, g), ENCARG(g)) ENCARG(cons_app(map, g)) -> c9(APP(map, g), ENCARG(map)) ENCARG(cons_app(nil, g)) -> c9(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(cons, g)) -> c9(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(filter, g)) -> c9(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(filter2, g)) -> c9(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(true, g)) -> c9(APP(true, g), ENCARG(true)) ENCARG(cons_app(false, g)) -> c9(APP(false, g), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), g)) -> c9(APP(app(encArg(z0), encArg(z1)), g), ENCARG(cons_app(z0, z1))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, g)) -> c9(APP(f, g), ENCARG(f)) ENCARG(cons_app(g, g)) -> c9(APP(g, g), ENCARG(g)) ENCARG(cons_app(map, g)) -> c9(APP(map, g), ENCARG(map)) ENCARG(cons_app(nil, g)) -> c9(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(cons, g)) -> c9(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(filter, g)) -> c9(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(filter2, g)) -> c9(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(true, g)) -> c9(APP(true, g), ENCARG(true)) ENCARG(cons_app(false, g)) -> c9(APP(false, g), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(filter2, g)) -> c9(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(true, g)) -> c9(APP(true, g), ENCARG(true)) ENCARG(cons_app(nil, g)) -> c9(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(g, g)) -> c9(APP(g, g), ENCARG(g)) ENCARG(cons_app(false, g)) -> c9(APP(false, g), ENCARG(false)) ENCARG(cons_app(filter, g)) -> c9(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(f, g)) -> c9(APP(f, g), ENCARG(f)) ENCARG(cons_app(cons, g)) -> c9(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(map, g)) -> c9(APP(map, g), ENCARG(map)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, map)) -> c9(APP(encArg(x0), map), ENCARG(x0)) by ENCARG(cons_app(f, map)) -> c9(APP(f, map), ENCARG(f)) ENCARG(cons_app(g, map)) -> c9(APP(g, map), ENCARG(g)) ENCARG(cons_app(map, map)) -> c9(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c9(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c9(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c9(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c9(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c9(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c9(APP(false, map), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), map)) -> c9(APP(app(encArg(z0), encArg(z1)), map), ENCARG(cons_app(z0, z1))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, map)) -> c9(APP(f, map), ENCARG(f)) ENCARG(cons_app(g, map)) -> c9(APP(g, map), ENCARG(g)) ENCARG(cons_app(map, map)) -> c9(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c9(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c9(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c9(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c9(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c9(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c9(APP(false, map), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(cons, map)) -> c9(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(true, map)) -> c9(APP(true, map), ENCARG(true)) ENCARG(cons_app(nil, map)) -> c9(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(false, map)) -> c9(APP(false, map), ENCARG(false)) ENCARG(cons_app(g, map)) -> c9(APP(g, map), ENCARG(g)) ENCARG(cons_app(f, map)) -> c9(APP(f, map), ENCARG(f)) ENCARG(cons_app(filter2, map)) -> c9(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(filter, map)) -> c9(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(map, map)) -> c9(APP(map, map), ENCARG(map)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) by ENCARG(cons_app(f, nil)) -> c9(APP(f, nil), ENCARG(f)) ENCARG(cons_app(g, nil)) -> c9(APP(g, nil), ENCARG(g)) ENCARG(cons_app(map, nil)) -> c9(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c9(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c9(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c9(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c9(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c9(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c9(APP(false, nil), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), nil)) -> c9(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, nil)) -> c9(APP(f, nil), ENCARG(f)) ENCARG(cons_app(g, nil)) -> c9(APP(g, nil), ENCARG(g)) ENCARG(cons_app(map, nil)) -> c9(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c9(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c9(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c9(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c9(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c9(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c9(APP(false, nil), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(nil, nil)) -> c9(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(filter2, nil)) -> c9(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(map, nil)) -> c9(APP(map, nil), ENCARG(map)) ENCARG(cons_app(g, nil)) -> c9(APP(g, nil), ENCARG(g)) ENCARG(cons_app(cons, nil)) -> c9(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(f, nil)) -> c9(APP(f, nil), ENCARG(f)) ENCARG(cons_app(true, nil)) -> c9(APP(true, nil), ENCARG(true)) ENCARG(cons_app(filter, nil)) -> c9(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(false, nil)) -> c9(APP(false, nil), ENCARG(false)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0)) by ENCARG(cons_app(f, cons)) -> c9(APP(f, cons), ENCARG(f)) ENCARG(cons_app(g, cons)) -> c9(APP(g, cons), ENCARG(g)) ENCARG(cons_app(map, cons)) -> c9(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c9(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c9(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c9(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c9(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c9(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c9(APP(false, cons), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), cons)) -> c9(APP(app(encArg(z0), encArg(z1)), cons), ENCARG(cons_app(z0, z1))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, cons)) -> c9(APP(f, cons), ENCARG(f)) ENCARG(cons_app(g, cons)) -> c9(APP(g, cons), ENCARG(g)) ENCARG(cons_app(map, cons)) -> c9(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c9(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c9(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c9(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c9(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c9(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c9(APP(false, cons), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(false, cons)) -> c9(APP(false, cons), ENCARG(false)) ENCARG(cons_app(filter2, cons)) -> c9(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(map, cons)) -> c9(APP(map, cons), ENCARG(map)) ENCARG(cons_app(filter, cons)) -> c9(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(g, cons)) -> c9(APP(g, cons), ENCARG(g)) ENCARG(cons_app(f, cons)) -> c9(APP(f, cons), ENCARG(f)) ENCARG(cons_app(true, cons)) -> c9(APP(true, cons), ENCARG(true)) ENCARG(cons_app(cons, cons)) -> c9(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(nil, cons)) -> c9(APP(nil, cons), ENCARG(nil)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0)) by ENCARG(cons_app(f, filter)) -> c9(APP(f, filter), ENCARG(f)) ENCARG(cons_app(g, filter)) -> c9(APP(g, filter), ENCARG(g)) ENCARG(cons_app(map, filter)) -> c9(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c9(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c9(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c9(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c9(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c9(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c9(APP(false, filter), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter)) -> c9(APP(app(encArg(z0), encArg(z1)), filter), ENCARG(cons_app(z0, z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, filter)) -> c9(APP(f, filter), ENCARG(f)) ENCARG(cons_app(g, filter)) -> c9(APP(g, filter), ENCARG(g)) ENCARG(cons_app(map, filter)) -> c9(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c9(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c9(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c9(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c9(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c9(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c9(APP(false, filter), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(f, filter)) -> c9(APP(f, filter), ENCARG(f)) ENCARG(cons_app(g, filter)) -> c9(APP(g, filter), ENCARG(g)) ENCARG(cons_app(filter2, filter)) -> c9(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(nil, filter)) -> c9(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(true, filter)) -> c9(APP(true, filter), ENCARG(true)) ENCARG(cons_app(cons, filter)) -> c9(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c9(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(false, filter)) -> c9(APP(false, filter), ENCARG(false)) ENCARG(cons_app(map, filter)) -> c9(APP(map, filter), ENCARG(map)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter2)) -> c9(APP(encArg(x0), filter2), ENCARG(x0)) by ENCARG(cons_app(f, filter2)) -> c9(APP(f, filter2), ENCARG(f)) ENCARG(cons_app(g, filter2)) -> c9(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(map, filter2)) -> c9(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c9(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c9(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c9(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c9(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c9(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c9(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter2)) -> c9(APP(app(encArg(z0), encArg(z1)), filter2), ENCARG(cons_app(z0, z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, filter2)) -> c9(APP(f, filter2), ENCARG(f)) ENCARG(cons_app(g, filter2)) -> c9(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(map, filter2)) -> c9(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c9(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c9(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c9(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c9(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c9(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c9(APP(false, filter2), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(filter2, filter2)) -> c9(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(cons, filter2)) -> c9(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c9(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(map, filter2)) -> c9(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(g, filter2)) -> c9(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(false, filter2)) -> c9(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(true, filter2)) -> c9(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(nil, filter2)) -> c9(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(f, filter2)) -> c9(APP(f, filter2), ENCARG(f)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0)) by ENCARG(cons_app(f, true)) -> c9(APP(f, true), ENCARG(f)) ENCARG(cons_app(g, true)) -> c9(APP(g, true), ENCARG(g)) ENCARG(cons_app(map, true)) -> c9(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c9(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c9(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c9(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c9(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c9(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c9(APP(false, true), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), true)) -> c9(APP(app(encArg(z0), encArg(z1)), true), ENCARG(cons_app(z0, z1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, true)) -> c9(APP(f, true), ENCARG(f)) ENCARG(cons_app(g, true)) -> c9(APP(g, true), ENCARG(g)) ENCARG(cons_app(map, true)) -> c9(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c9(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c9(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c9(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c9(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c9(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c9(APP(false, true), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(false, true)) -> c9(APP(false, true), ENCARG(false)) ENCARG(cons_app(filter, true)) -> c9(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c9(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(f, true)) -> c9(APP(f, true), ENCARG(f)) ENCARG(cons_app(g, true)) -> c9(APP(g, true), ENCARG(g)) ENCARG(cons_app(cons, true)) -> c9(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(true, true)) -> c9(APP(true, true), ENCARG(true)) ENCARG(cons_app(nil, true)) -> c9(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(map, true)) -> c9(APP(map, true), ENCARG(map)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0)) by ENCARG(cons_app(f, false)) -> c9(APP(f, false), ENCARG(f)) ENCARG(cons_app(g, false)) -> c9(APP(g, false), ENCARG(g)) ENCARG(cons_app(map, false)) -> c9(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c9(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c9(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c9(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c9(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c9(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c9(APP(false, false), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), false)) -> c9(APP(app(encArg(z0), encArg(z1)), false), ENCARG(cons_app(z0, z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, false)) -> c9(APP(f, false), ENCARG(f)) ENCARG(cons_app(g, false)) -> c9(APP(g, false), ENCARG(g)) ENCARG(cons_app(map, false)) -> c9(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c9(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c9(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c9(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c9(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c9(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c9(APP(false, false), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(cons, false)) -> c9(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(g, false)) -> c9(APP(g, false), ENCARG(g)) ENCARG(cons_app(true, false)) -> c9(APP(true, false), ENCARG(true)) ENCARG(cons_app(filter2, false)) -> c9(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(nil, false)) -> c9(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(false, false)) -> c9(APP(false, false), ENCARG(false)) ENCARG(cons_app(f, false)) -> c9(APP(f, false), ENCARG(f)) ENCARG(cons_app(map, false)) -> c9(APP(map, false), ENCARG(map)) ENCARG(cons_app(filter, false)) -> c9(APP(filter, false), ENCARG(filter)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(f, x1)) -> c9(APP(f, encArg(x1)), ENCARG(x1)) by ENCARG(cons_app(f, f)) -> c9(APP(f, f), ENCARG(f)) ENCARG(cons_app(f, g)) -> c9(APP(f, g), ENCARG(g)) ENCARG(cons_app(f, map)) -> c9(APP(f, map), ENCARG(map)) ENCARG(cons_app(f, nil)) -> c9(APP(f, nil), ENCARG(nil)) ENCARG(cons_app(f, cons)) -> c9(APP(f, cons), ENCARG(cons)) ENCARG(cons_app(f, filter)) -> c9(APP(f, filter), ENCARG(filter)) ENCARG(cons_app(f, filter2)) -> c9(APP(f, filter2), ENCARG(filter2)) ENCARG(cons_app(f, true)) -> c9(APP(f, true), ENCARG(true)) ENCARG(cons_app(f, false)) -> c9(APP(f, false), ENCARG(false)) ENCARG(cons_app(f, cons_app(z0, z1))) -> c9(APP(f, app(encArg(z0), encArg(z1))), ENCARG(cons_app(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, f)) -> c9(APP(f, f), ENCARG(f)) ENCARG(cons_app(f, g)) -> c9(APP(f, g), ENCARG(g)) ENCARG(cons_app(f, map)) -> c9(APP(f, map), ENCARG(map)) ENCARG(cons_app(f, nil)) -> c9(APP(f, nil), ENCARG(nil)) ENCARG(cons_app(f, cons)) -> c9(APP(f, cons), ENCARG(cons)) ENCARG(cons_app(f, filter)) -> c9(APP(f, filter), ENCARG(filter)) ENCARG(cons_app(f, filter2)) -> c9(APP(f, filter2), ENCARG(filter2)) ENCARG(cons_app(f, true)) -> c9(APP(f, true), ENCARG(true)) ENCARG(cons_app(f, false)) -> c9(APP(f, false), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(f, f)) -> c9(APP(f, f), ENCARG(f)) ENCARG(cons_app(f, filter2)) -> c9(APP(f, filter2), ENCARG(filter2)) ENCARG(cons_app(f, g)) -> c9(APP(f, g), ENCARG(g)) ENCARG(cons_app(f, true)) -> c9(APP(f, true), ENCARG(true)) ENCARG(cons_app(f, nil)) -> c9(APP(f, nil), ENCARG(nil)) ENCARG(cons_app(f, map)) -> c9(APP(f, map), ENCARG(map)) ENCARG(cons_app(f, cons)) -> c9(APP(f, cons), ENCARG(cons)) ENCARG(cons_app(f, false)) -> c9(APP(f, false), ENCARG(false)) ENCARG(cons_app(f, filter)) -> c9(APP(f, filter), ENCARG(filter)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_2, c9_1, c25_2, c25_1, c_1, c9_3 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(g, x1)) -> c9(APP(g, encArg(x1)), ENCARG(x1)) by ENCARG(cons_app(g, f)) -> c9(APP(g, f), ENCARG(f)) ENCARG(cons_app(g, g)) -> c9(APP(g, g), ENCARG(g)) ENCARG(cons_app(g, map)) -> c9(APP(g, map), ENCARG(map)) ENCARG(cons_app(g, nil)) -> c9(APP(g, nil), ENCARG(nil)) ENCARG(cons_app(g, cons)) -> c9(APP(g, cons), ENCARG(cons)) ENCARG(cons_app(g, filter)) -> c9(APP(g, filter), ENCARG(filter)) ENCARG(cons_app(g, filter2)) -> c9(APP(g, filter2), ENCARG(filter2)) ENCARG(cons_app(g, true)) -> c9(APP(g, true), ENCARG(true)) ENCARG(cons_app(g, false)) -> c9(APP(g, false), ENCARG(false)) ENCARG(cons_app(g, cons_app(z0, z1))) -> c9(APP(g, app(encArg(z0), encArg(z1))), ENCARG(cons_app(z0, z1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, f)) -> c9(APP(g, f), ENCARG(f)) ENCARG(cons_app(g, g)) -> c9(APP(g, g), ENCARG(g)) ENCARG(cons_app(g, map)) -> c9(APP(g, map), ENCARG(map)) ENCARG(cons_app(g, nil)) -> c9(APP(g, nil), ENCARG(nil)) ENCARG(cons_app(g, cons)) -> c9(APP(g, cons), ENCARG(cons)) ENCARG(cons_app(g, filter)) -> c9(APP(g, filter), ENCARG(filter)) ENCARG(cons_app(g, filter2)) -> c9(APP(g, filter2), ENCARG(filter2)) ENCARG(cons_app(g, true)) -> c9(APP(g, true), ENCARG(true)) ENCARG(cons_app(g, false)) -> c9(APP(g, false), ENCARG(false)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_2, c25_1, c_1, c9_3, c9_2 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(g, cons)) -> c9(APP(g, cons), ENCARG(cons)) ENCARG(cons_app(g, f)) -> c9(APP(g, f), ENCARG(f)) ENCARG(cons_app(g, true)) -> c9(APP(g, true), ENCARG(true)) ENCARG(cons_app(g, nil)) -> c9(APP(g, nil), ENCARG(nil)) ENCARG(cons_app(g, map)) -> c9(APP(g, map), ENCARG(map)) ENCARG(cons_app(g, filter2)) -> c9(APP(g, filter2), ENCARG(filter2)) ENCARG(cons_app(g, false)) -> c9(APP(g, false), ENCARG(false)) ENCARG(cons_app(g, filter)) -> c9(APP(g, filter), ENCARG(filter)) ENCARG(cons_app(g, g)) -> c9(APP(g, g), ENCARG(g)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_2, c25_1, c_1, c9_3, c9_2 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c25(APP(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2), APP(g, app(g, z0))) by APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(app(app(app(filter2, app(g, app(f, z0))), g), app(g, app(f, z0))), x1), APP(g, app(g, app(f, z0)))) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(app(app(app(filter2, app(g, app(f, z0))), g), app(g, app(f, z0))), x1), APP(g, app(g, app(f, z0)))) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(app(app(app(filter2, app(g, app(f, z0))), g), app(g, app(f, z0))), x1), APP(g, app(g, app(f, z0)))) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_2, c25_1, c_1, c9_3, c9_2 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_2, c25_1, c_1, c9_3, c9_2 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(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)) -> c25(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)) -> c25(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, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(g, app(f, z0))), f), app(f, z0)), x2)), app(filter, f)), app(app(cons, app(f, z0)), x2)), x3), APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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)) -> c25(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, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(g, app(f, z0))), f), app(f, z0)), x2)), app(filter, f)), app(app(cons, app(f, z0)), x2)), x3), APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, true), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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)) -> c25(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, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(g, app(f, z0))), f), app(f, z0)), x2)), app(filter, f)), app(app(cons, app(f, z0)), x2)), x3), APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_2, c25_1, c_1, c9_3, c9_2 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c25(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, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_2, c25_1, c_1, c9_3, c9_2 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) by ENCODE_APP(f, f) -> c(APP(f, f)) ENCODE_APP(g, f) -> c(APP(g, f)) ENCODE_APP(map, f) -> c(APP(map, f)) ENCODE_APP(nil, f) -> c(APP(nil, f)) ENCODE_APP(cons, f) -> c(APP(cons, f)) ENCODE_APP(filter, f) -> c(APP(filter, f)) ENCODE_APP(filter2, f) -> c(APP(filter2, f)) ENCODE_APP(true, f) -> c(APP(true, f)) ENCODE_APP(false, f) -> c(APP(false, f)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(f, f) -> c(APP(f, f)) ENCODE_APP(g, f) -> c(APP(g, f)) ENCODE_APP(map, f) -> c(APP(map, f)) ENCODE_APP(nil, f) -> c(APP(nil, f)) ENCODE_APP(cons, f) -> c(APP(cons, f)) ENCODE_APP(filter, f) -> c(APP(filter, f)) ENCODE_APP(filter2, f) -> c(APP(filter2, f)) ENCODE_APP(true, f) -> c(APP(true, f)) ENCODE_APP(false, f) -> c(APP(false, f)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(true, f) -> c(APP(true, f)) ENCODE_APP(filter2, f) -> c(APP(filter2, f)) ENCODE_APP(f, f) -> c(APP(f, f)) ENCODE_APP(filter, f) -> c(APP(filter, f)) ENCODE_APP(false, f) -> c(APP(false, f)) ENCODE_APP(map, f) -> c(APP(map, f)) ENCODE_APP(g, f) -> c(APP(g, f)) ENCODE_APP(cons, f) -> c(APP(cons, f)) ENCODE_APP(nil, f) -> c(APP(nil, f)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) 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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) by ENCODE_APP(f, g) -> c(APP(f, g)) ENCODE_APP(g, g) -> c(APP(g, 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)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(f, g) -> c(APP(f, g)) ENCODE_APP(g, g) -> c(APP(g, 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(filter, g) -> c(APP(filter, g)) ENCODE_APP(map, g) -> c(APP(map, g)) ENCODE_APP(true, g) -> c(APP(true, g)) ENCODE_APP(filter2, g) -> c(APP(filter2, g)) ENCODE_APP(nil, g) -> c(APP(nil, g)) ENCODE_APP(false, g) -> c(APP(false, g)) ENCODE_APP(f, g) -> c(APP(f, g)) ENCODE_APP(cons, g) -> c(APP(cons, g)) ENCODE_APP(g, g) -> c(APP(g, g)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) by ENCODE_APP(f, map) -> c(APP(f, map)) ENCODE_APP(g, map) -> c(APP(g, 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)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(f, map) -> c(APP(f, map)) ENCODE_APP(g, map) -> c(APP(g, 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(filter, map) -> c(APP(filter, map)) ENCODE_APP(true, map) -> c(APP(true, map)) ENCODE_APP(f, map) -> c(APP(f, map)) ENCODE_APP(filter2, map) -> c(APP(filter2, map)) ENCODE_APP(nil, map) -> c(APP(nil, map)) ENCODE_APP(false, map) -> c(APP(false, map)) ENCODE_APP(cons, map) -> c(APP(cons, map)) ENCODE_APP(g, map) -> c(APP(g, map)) ENCODE_APP(map, map) -> c(APP(map, map)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) by ENCODE_APP(f, nil) -> c(APP(f, nil)) ENCODE_APP(g, nil) -> c(APP(g, 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)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(f, nil) -> c(APP(f, nil)) ENCODE_APP(g, nil) -> c(APP(g, 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (119) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(map, nil) -> c(APP(map, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(filter2, nil) -> c(APP(filter2, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(g, nil) -> c(APP(g, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(f, nil) -> c(APP(f, nil)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(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(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) by ENCODE_APP(f, cons) -> c(APP(f, cons)) ENCODE_APP(g, cons) -> c(APP(g, cons)) ENCODE_APP(map, cons) -> c(APP(map, cons)) ENCODE_APP(nil, cons) -> c(APP(nil, cons)) ENCODE_APP(cons, cons) -> c(APP(cons, cons)) ENCODE_APP(filter, cons) -> c(APP(filter, cons)) ENCODE_APP(filter2, cons) -> c(APP(filter2, cons)) ENCODE_APP(true, cons) -> c(APP(true, cons)) ENCODE_APP(false, cons) -> c(APP(false, cons)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(f, cons) -> c(APP(f, cons)) ENCODE_APP(g, cons) -> c(APP(g, cons)) ENCODE_APP(map, cons) -> c(APP(map, cons)) ENCODE_APP(nil, cons) -> c(APP(nil, cons)) ENCODE_APP(cons, cons) -> c(APP(cons, cons)) ENCODE_APP(filter, cons) -> c(APP(filter, cons)) ENCODE_APP(filter2, cons) -> c(APP(filter2, cons)) ENCODE_APP(true, cons) -> c(APP(true, cons)) ENCODE_APP(false, cons) -> c(APP(false, cons)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (123) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(map, cons) -> c(APP(map, cons)) ENCODE_APP(f, cons) -> c(APP(f, cons)) ENCODE_APP(nil, cons) -> c(APP(nil, cons)) ENCODE_APP(true, cons) -> c(APP(true, cons)) ENCODE_APP(filter2, cons) -> c(APP(filter2, cons)) ENCODE_APP(g, cons) -> c(APP(g, cons)) ENCODE_APP(filter, cons) -> c(APP(filter, cons)) ENCODE_APP(false, cons) -> c(APP(false, cons)) ENCODE_APP(cons, cons) -> c(APP(cons, cons)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) by ENCODE_APP(f, filter) -> c(APP(f, filter)) ENCODE_APP(g, filter) -> c(APP(g, filter)) ENCODE_APP(map, filter) -> c(APP(map, filter)) ENCODE_APP(nil, filter) -> c(APP(nil, filter)) ENCODE_APP(cons, filter) -> c(APP(cons, filter)) ENCODE_APP(filter, filter) -> c(APP(filter, filter)) ENCODE_APP(filter2, filter) -> c(APP(filter2, filter)) ENCODE_APP(true, filter) -> c(APP(true, filter)) ENCODE_APP(false, filter) -> c(APP(false, filter)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(f, filter) -> c(APP(f, filter)) ENCODE_APP(g, filter) -> c(APP(g, filter)) ENCODE_APP(map, filter) -> c(APP(map, filter)) ENCODE_APP(nil, filter) -> c(APP(nil, filter)) ENCODE_APP(cons, filter) -> c(APP(cons, filter)) ENCODE_APP(filter, filter) -> c(APP(filter, filter)) ENCODE_APP(filter2, filter) -> c(APP(filter2, filter)) ENCODE_APP(true, filter) -> c(APP(true, filter)) ENCODE_APP(false, filter) -> c(APP(false, filter)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (127) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(g, filter) -> c(APP(g, filter)) ENCODE_APP(false, filter) -> c(APP(false, filter)) ENCODE_APP(true, filter) -> c(APP(true, filter)) ENCODE_APP(cons, filter) -> c(APP(cons, filter)) ENCODE_APP(nil, filter) -> c(APP(nil, filter)) ENCODE_APP(f, filter) -> c(APP(f, filter)) ENCODE_APP(filter, filter) -> c(APP(filter, filter)) ENCODE_APP(map, filter) -> c(APP(map, filter)) ENCODE_APP(filter2, filter) -> c(APP(filter2, filter)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(g) -> g 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(f, app(f, z0)) -> app(g, app(f, z0)) app(g, app(g, z0)) -> app(f, z0) 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(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) ENCARG(cons_app(map, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c9(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c9(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c9(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c9(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(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)))) -> c9(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(f, x2))) -> c9(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c9(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c9(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c9(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(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))) -> c9(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))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c9(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c9(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c9(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c9(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c9(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c9(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(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)) -> c9(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(f, x1), x2)) -> c9(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c9(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c9(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(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)) -> c9(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), f)) -> c9(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c9(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c9(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c9(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c9(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(APP(app(app(app(filter2, false), z0), x1), nil)) ENCODE_APP(cons_app(z0, z1), f) -> c(APP(app(encArg(z0), encArg(z1)), f)) ENCODE_APP(cons_app(z0, z1), g) -> c(APP(app(encArg(z0), encArg(z1)), g)) ENCODE_APP(cons_app(z0, z1), map) -> c(APP(app(encArg(z0), encArg(z1)), map)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), filter) -> c(APP(app(encArg(z0), encArg(z1)), filter)) S tuples: APP(g, app(g, z0)) -> c21(APP(f, z0)) APP(app(map, z0), nil) -> c22 APP(app(filter, z0), nil) -> c24 APP(f, app(f, z0)) -> c20(APP(f, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c23(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c26(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c27(APP(app(filter, z0), z2)) APP(app(filter, f), app(app(cons, app(f, z0)), x2)) -> c25(APP(f, app(f, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c25(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c25(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c25(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c25(APP(app(app(app(filter2, true), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(g, x0)), x1)) -> c25(APP(g, app(g, x0))) APP(app(filter, g), app(app(cons, app(g, app(f, z0))), x1)) -> c25(APP(g, app(g, app(f, z0)))) APP(app(filter, app(filter, app(app(app(filter2, false), z0), z1))), app(app(cons, app(app(cons, z2), x2)), x3)) -> c25(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, g)), app(app(cons, app(app(cons, app(g, z0)), x2)), x3)) -> c25(APP(app(app(app(filter2, app(app(app(app(filter2, app(f, z0)), g), app(g, z0)), x2)), app(filter, g)), app(app(cons, app(g, z0)), x2)), x3), APP(app(filter, g), app(app(cons, app(g, z0)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c25(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)) -> c25(APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2))) APP(app(filter, app(filter, f)), app(app(cons, app(app(cons, app(f, z0)), x2)), x3)) -> c25(APP(app(filter, f), app(app(cons, app(f, z0)), x2))) APP(app(filter, app(filter, app(map, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(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)) -> c25(APP(app(app(app(filter2, false), x0), x1), x2)) APP(app(filter, app(app(app(filter2, false), z0), x1)), app(app(cons, nil), x3)) -> c25(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: c21_1, c22, c24, c20_1, c23_2, c26_1, c27_1, c9_1, c25_1, c_1, c9_3, c9_2, c25_2 ---------------------------------------- (129) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (130) 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(f, c_app(f, x)) -> app(g, app(f, x)) app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) app(c_app(map, fun), nil) -> nil app(c_app(filter, fun), nil) -> nil app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(g, c_app(g, x)) -> app(f, x) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(g) -> g 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_f -> f encode_g -> g 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 ---------------------------------------- (131) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (132) 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(f, c_app(f, x)) -> app(g, app(f, x)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(g, c_app(g, x)) -> app(f, x) [1] encArg(f) -> f [0] encArg(g) -> g [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_f -> f [0] encode_g -> g [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 ---------------------------------------- (133) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (134) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(f, c_app(f, x)) -> app(g, app(f, x)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(g, c_app(g, x)) -> app(f, x) [1] encArg(f) -> f [0] encArg(g) -> g [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_f -> f [0] encode_g -> g [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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app c_app :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app g :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app map :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app cons :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app filter2 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app true :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app filter :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app false :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app nil :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encArg :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app cons_app :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_app :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_g :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_map :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_nil :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_cons :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_filter :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_filter2 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_true :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app encode_false :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app Rewrite Strategy: INNERMOST ---------------------------------------- (135) 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_f encode_g 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_f -> null_encode_f [0] encode_g -> null_encode_g [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_f, null_encode_g, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app ---------------------------------------- (136) 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(f, c_app(f, x)) -> app(g, app(f, x)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(g, c_app(g, x)) -> app(f, x) [1] encArg(f) -> f [0] encArg(g) -> g [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_f -> f [0] encode_g -> g [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_f -> null_encode_f [0] encode_g -> null_encode_g [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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app g :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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_f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 ---------------------------------------- (137) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (138) 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(f, c_app(f, c_app(f, x'))) -> app(g, app(g, app(f, x'))) [2] app(f, c_app(f, x)) -> app(g, c_app(f, x)) [1] app(f, c_app(f, x)) -> app(g, null_app) [1] app(c_app(map, f), c_app(c_app(cons, c_app(f, x'')), xs)) -> app(app(cons, app(g, app(f, x''))), app(c_app(map, f), xs)) [2] app(c_app(map, f), c_app(c_app(cons, c_app(f, x'')), xs)) -> app(app(cons, app(g, app(f, x''))), app(null_app, xs)) [2] app(c_app(map, c_app(map, fun')), c_app(c_app(cons, c_app(c_app(cons, x2), xs')), xs)) -> app(app(cons, app(app(cons, app(fun', x2)), app(app(map, fun'), xs'))), app(c_app(map, c_app(map, fun')), xs)) [2] app(c_app(map, c_app(map, fun')), c_app(c_app(cons, c_app(c_app(cons, x2), xs')), xs)) -> app(app(cons, app(app(cons, app(fun', x2)), app(app(map, fun'), xs'))), app(null_app, xs)) [2] app(c_app(map, c_app(c_app(c_app(filter2, true), fun''), x3)), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(cons, x3), app(app(filter, fun''), x))), app(c_app(map, c_app(c_app(c_app(filter2, true), fun''), x3)), xs)) [2] app(c_app(map, c_app(c_app(c_app(filter2, true), fun''), x3)), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(cons, x3), app(app(filter, fun''), x))), app(null_app, xs)) [2] app(c_app(map, c_app(c_app(c_app(filter2, false), fun1), x4)), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(filter, fun1), x)), app(c_app(map, c_app(c_app(c_app(filter2, false), fun1), x4)), xs)) [2] app(c_app(map, c_app(c_app(c_app(filter2, false), fun1), x4)), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(filter, fun1), x)), app(null_app, xs)) [2] app(c_app(map, c_app(map, fun2)), c_app(c_app(cons, nil), xs)) -> app(app(cons, nil), app(c_app(map, c_app(map, fun2)), xs)) [2] app(c_app(map, c_app(map, fun2)), c_app(c_app(cons, nil), xs)) -> app(app(cons, nil), app(null_app, xs)) [2] app(c_app(map, c_app(filter, fun3)), c_app(c_app(cons, nil), xs)) -> app(app(cons, nil), app(c_app(map, c_app(filter, fun3)), xs)) [2] app(c_app(map, c_app(filter, fun3)), c_app(c_app(cons, nil), xs)) -> app(app(cons, nil), app(null_app, xs)) [2] app(c_app(map, c_app(filter, fun4)), c_app(c_app(cons, c_app(c_app(cons, x5), xs'')), xs)) -> app(app(cons, app(app(app(app(filter2, app(fun4, x5)), fun4), x5), xs'')), app(c_app(map, c_app(filter, fun4)), xs)) [2] app(c_app(map, c_app(filter, fun4)), c_app(c_app(cons, c_app(c_app(cons, x5), xs'')), xs)) -> app(app(cons, app(app(app(app(filter2, app(fun4, x5)), fun4), x5), xs'')), app(null_app, xs)) [2] app(c_app(map, g), c_app(c_app(cons, c_app(g, x6)), xs)) -> app(app(cons, app(f, x6)), app(c_app(map, g), xs)) [2] app(c_app(map, g), c_app(c_app(cons, c_app(g, x6)), xs)) -> app(app(cons, app(f, x6)), app(null_app, xs)) [2] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, c_app(fun, x)), app(c_app(map, fun), xs)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, c_app(fun, x)), app(null_app, xs)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, null_app), app(c_app(map, fun), xs)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, null_app), app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(c_app(cons, x), app(c_app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(c_app(cons, x), app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(null_app, app(c_app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(null_app, app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(c_app(filter, fun), xs) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(null_app, xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, f), c_app(c_app(cons, c_app(f, x7)), xs)) -> app(app(app(app(filter2, app(g, app(f, x7))), f), c_app(f, x7)), xs) [2] app(c_app(filter, c_app(map, fun5)), c_app(c_app(cons, c_app(c_app(cons, x8), xs1)), xs)) -> app(app(app(app(filter2, app(app(cons, app(fun5, x8)), app(app(map, fun5), xs1))), c_app(map, fun5)), c_app(c_app(cons, x8), xs1)), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, true), fun6), x9)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(cons, x9), app(app(filter, fun6), x))), c_app(c_app(c_app(filter2, true), fun6), x9)), x), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, false), fun7), x10)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(filter, fun7), x)), c_app(c_app(c_app(filter2, false), fun7), x10)), x), xs) [2] app(c_app(filter, c_app(map, fun8)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(map, fun8)), nil), xs) [2] app(c_app(filter, c_app(filter, fun9)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(filter, fun9)), nil), xs) [2] app(c_app(filter, c_app(filter, fun10)), c_app(c_app(cons, c_app(c_app(cons, x11), xs2)), xs)) -> app(app(app(app(filter2, app(app(app(app(filter2, app(fun10, x11)), fun10), x11), xs2)), c_app(filter, fun10)), c_app(c_app(cons, x11), xs2)), xs) [2] app(c_app(filter, g), c_app(c_app(cons, c_app(g, x12)), xs)) -> app(app(app(app(filter2, app(f, x12)), g), c_app(g, x12)), xs) [2] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, c_app(fun, x)), fun), x), xs) [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, null_app), fun), x), xs) [1] app(g, c_app(g, x)) -> app(f, x) [1] encArg(f) -> f [0] encArg(g) -> g [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_f -> f [0] encode_g -> g [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_f -> null_encode_f [0] encode_g -> null_encode_g [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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app g :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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_f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 ---------------------------------------- (139) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: f => 1 g => 5 map => 6 cons => 0 filter2 => 4 true => 8 filter => 3 false => 2 nil => 7 null_encArg => 0 null_encode_app => 0 null_encode_f => 0 null_encode_g => 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 ---------------------------------------- (140) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 2 }-> app(app(app(app(4, app(app(app(app(4, app(fun10, x11)), fun10), x11), xs2)), 1 + 3 + fun10), 1 + (1 + 0 + x11) + xs2), xs) :|: z = 1 + 3 + (1 + 3 + fun10), xs >= 0, xs2 >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x11) + xs2)) + xs, x11 >= 0, fun10 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, app(app(3, fun7), x)), 1 + (1 + (1 + 4 + 2) + fun7) + x10), x), xs) :|: z = 1 + 3 + (1 + (1 + (1 + 4 + 2) + fun7) + x10), xs >= 0, x >= 0, x10 >= 0, z' = 1 + (1 + 0 + x) + xs, fun7 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, app(app(0, x9), app(app(3, fun6), x))), 1 + (1 + (1 + 4 + 8) + fun6) + x9), x), xs) :|: z = 1 + 3 + (1 + (1 + (1 + 4 + 8) + fun6) + x9), fun6 >= 0, xs >= 0, x >= 0, x9 >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(app(app(4, app(app(0, app(fun5, x8)), app(app(6, fun5), xs1))), 1 + 6 + fun5), 1 + (1 + 0 + x8) + xs1), xs) :|: z' = 1 + (1 + 0 + (1 + (1 + 0 + x8) + xs1)) + xs, z = 1 + 3 + (1 + 6 + fun5), xs >= 0, x8 >= 0, xs1 >= 0, fun5 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, app(5, app(1, x7))), 1), 1 + 1 + x7), xs) :|: z = 1 + 3 + 1, xs >= 0, z' = 1 + (1 + 0 + (1 + 1 + x7)) + xs, x7 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, app(1, x12)), 5), 1 + 5 + x12), xs) :|: xs >= 0, x12 >= 0, z = 1 + 3 + 5, z' = 1 + (1 + 0 + (1 + 5 + x12)) + xs app(z, z') -{ 2 }-> app(app(app(app(4, 7), 1 + 6 + fun8), 7), xs) :|: xs >= 0, z = 1 + 3 + (1 + 6 + fun8), z' = 1 + (1 + 0 + 7) + xs, fun8 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, 7), 1 + 3 + fun9), 7), xs) :|: xs >= 0, z' = 1 + (1 + 0 + 7) + xs, z = 1 + 3 + (1 + 3 + fun9), fun9 >= 0 app(z, z') -{ 1 }-> app(app(app(app(4, 0), fun), x), xs) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 3 + fun, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(app(app(app(4, 1 + fun + x), fun), x), xs) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 3 + fun, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(app(app(4, app(fun4, x5)), fun4), x5), xs'')), app(0, xs)) :|: xs >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x5) + xs'')) + xs, x5 >= 0, xs'' >= 0, fun4 >= 0, z = 1 + 6 + (1 + 3 + fun4) app(z, z') -{ 2 }-> app(app(0, app(app(app(app(4, app(fun4, x5)), fun4), x5), xs'')), app(1 + 6 + (1 + 3 + fun4), xs)) :|: xs >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x5) + xs'')) + xs, x5 >= 0, xs'' >= 0, fun4 >= 0, z = 1 + 6 + (1 + 3 + fun4) app(z, z') -{ 2 }-> app(app(0, app(app(3, fun1), x)), app(0, xs)) :|: xs >= 0, x4 >= 0, z = 1 + 6 + (1 + (1 + (1 + 4 + 2) + fun1) + x4), x >= 0, fun1 >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(3, fun1), x)), app(1 + 6 + (1 + (1 + (1 + 4 + 2) + fun1) + x4), xs)) :|: xs >= 0, x4 >= 0, z = 1 + 6 + (1 + (1 + (1 + 4 + 2) + fun1) + x4), x >= 0, fun1 >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(0, x3), app(app(3, fun''), x))), app(0, xs)) :|: xs >= 0, x >= 0, fun'' >= 0, x3 >= 0, z = 1 + 6 + (1 + (1 + (1 + 4 + 8) + fun'') + x3), z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(0, x3), app(app(3, fun''), x))), app(1 + 6 + (1 + (1 + (1 + 4 + 8) + fun'') + x3), xs)) :|: xs >= 0, x >= 0, fun'' >= 0, x3 >= 0, z = 1 + 6 + (1 + (1 + (1 + 4 + 8) + fun'') + x3), z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(0, app(fun', x2)), app(app(6, fun'), xs'))), app(0, xs)) :|: xs >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x2) + xs')) + xs, xs' >= 0, z = 1 + 6 + (1 + 6 + fun'), fun' >= 0, x2 >= 0 app(z, z') -{ 2 }-> app(app(0, app(app(0, app(fun', x2)), app(app(6, fun'), xs'))), app(1 + 6 + (1 + 6 + fun'), xs)) :|: xs >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x2) + xs')) + xs, xs' >= 0, z = 1 + 6 + (1 + 6 + fun'), fun' >= 0, x2 >= 0 app(z, z') -{ 2 }-> app(app(0, app(5, app(1, x''))), app(0, xs)) :|: xs >= 0, z = 1 + 6 + 1, x'' >= 0, z' = 1 + (1 + 0 + (1 + 1 + x'')) + xs app(z, z') -{ 2 }-> app(app(0, app(5, app(1, x''))), app(1 + 6 + 1, xs)) :|: xs >= 0, z = 1 + 6 + 1, x'' >= 0, z' = 1 + (1 + 0 + (1 + 1 + x'')) + xs app(z, z') -{ 2 }-> app(app(0, app(1, x6)), app(0, xs)) :|: z' = 1 + (1 + 0 + (1 + 5 + x6)) + xs, xs >= 0, z = 1 + 6 + 5, x6 >= 0 app(z, z') -{ 2 }-> app(app(0, app(1, x6)), app(1 + 6 + 5, xs)) :|: z' = 1 + (1 + 0 + (1 + 5 + x6)) + xs, xs >= 0, z = 1 + 6 + 5, x6 >= 0 app(z, z') -{ 2 }-> app(app(0, 7), app(0, xs)) :|: fun2 >= 0, xs >= 0, z' = 1 + (1 + 0 + 7) + xs, z = 1 + 6 + (1 + 6 + fun2) app(z, z') -{ 2 }-> app(app(0, 7), app(0, xs)) :|: z = 1 + 6 + (1 + 3 + fun3), xs >= 0, z' = 1 + (1 + 0 + 7) + xs, fun3 >= 0 app(z, z') -{ 2 }-> app(app(0, 7), app(1 + 6 + (1 + 6 + fun2), xs)) :|: fun2 >= 0, xs >= 0, z' = 1 + (1 + 0 + 7) + xs, z = 1 + 6 + (1 + 6 + fun2) app(z, z') -{ 2 }-> app(app(0, 7), app(1 + 6 + (1 + 3 + fun3), xs)) :|: z = 1 + 6 + (1 + 3 + fun3), xs >= 0, z' = 1 + (1 + 0 + 7) + xs, fun3 >= 0 app(z, z') -{ 1 }-> app(app(0, 0), app(0, xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 6 + fun, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(app(0, 0), app(1 + 6 + fun, xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 6 + fun, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(app(0, 1 + fun + x), app(0, xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 6 + fun, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(app(0, 1 + fun + x), app(1 + 6 + fun, xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 6 + fun, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(5, app(5, app(1, x'))) :|: z = 1, x' >= 0, z' = 1 + 1 + (1 + 1 + x') app(z, z') -{ 1 }-> app(5, 0) :|: z = 1, z' = 1 + 1 + x, x >= 0 app(z, z') -{ 1 }-> app(5, 1 + 1 + x) :|: z = 1, z' = 1 + 1 + x, x >= 0 app(z, z') -{ 1 }-> app(1, x) :|: z' = 1 + 5 + x, x >= 0, z = 5 app(z, z') -{ 1 }-> app(0, xs) :|: z = 1 + (1 + (1 + 4 + 2) + fun) + x, fun >= 0, xs >= 0, x >= 0, z' = xs app(z, z') -{ 1 }-> app(0, app(0, xs)) :|: fun >= 0, xs >= 0, z = 1 + (1 + (1 + 4 + 8) + fun) + x, x >= 0, z' = xs app(z, z') -{ 1 }-> app(0, app(1 + 3 + fun, xs)) :|: fun >= 0, xs >= 0, z = 1 + (1 + (1 + 4 + 8) + fun) + x, x >= 0, z' = xs app(z, z') -{ 1 }-> app(1 + 3 + fun, xs) :|: z = 1 + (1 + (1 + 4 + 2) + fun) + x, fun >= 0, xs >= 0, x >= 0, z' = xs app(z, z') -{ 1 }-> app(1 + 0 + x, app(0, xs)) :|: fun >= 0, xs >= 0, z = 1 + (1 + (1 + 4 + 8) + fun) + x, x >= 0, z' = xs app(z, z') -{ 1 }-> app(1 + 0 + x, app(1 + 3 + fun, xs)) :|: fun >= 0, xs >= 0, z = 1 + (1 + (1 + 4 + 8) + fun) + x, x >= 0, z' = xs app(z, z') -{ 1 }-> 7 :|: fun >= 0, z' = 7, z = 1 + 6 + fun app(z, z') -{ 1 }-> 7 :|: fun >= 0, z' = 7, z = 1 + 3 + fun app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 8 :|: z = 8 encArg(z) -{ 0 }-> 7 :|: z = 7 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_cons -{ 0 }-> 0 :|: encode_f -{ 0 }-> 1 :|: encode_f -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 3 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 4 :|: encode_filter2 -{ 0 }-> 0 :|: encode_g -{ 0 }-> 5 :|: encode_g -{ 0 }-> 0 :|: encode_map -{ 0 }-> 6 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 7 :|: encode_nil -{ 0 }-> 0 :|: encode_true -{ 0 }-> 8 :|: encode_true -{ 0 }-> 0 :|: ---------------------------------------- (141) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (142) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 2 }-> app(app(app(app(4, app(app(app(app(4, app(z - 8, x11)), z - 8), x11), xs2)), 1 + 3 + (z - 8)), 1 + (1 + 0 + x11) + xs2), xs) :|: xs >= 0, xs2 >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x11) + xs2)) + xs, x11 >= 0, z - 8 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, app(app(3, fun7), x)), 1 + (1 + (1 + 4 + 2) + fun7) + x10), x), xs) :|: z = 1 + 3 + (1 + (1 + (1 + 4 + 2) + fun7) + x10), xs >= 0, x >= 0, x10 >= 0, z' = 1 + (1 + 0 + x) + xs, fun7 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, app(app(0, x9), app(app(3, fun6), x))), 1 + (1 + (1 + 4 + 8) + fun6) + x9), x), xs) :|: z = 1 + 3 + (1 + (1 + (1 + 4 + 8) + fun6) + x9), fun6 >= 0, xs >= 0, x >= 0, x9 >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(app(app(4, app(app(0, app(z - 11, x8)), app(app(6, z - 11), xs1))), 1 + 6 + (z - 11)), 1 + (1 + 0 + x8) + xs1), xs) :|: z' = 1 + (1 + 0 + (1 + (1 + 0 + x8) + xs1)) + xs, xs >= 0, x8 >= 0, xs1 >= 0, z - 11 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, app(5, app(1, x7))), 1), 1 + 1 + x7), xs) :|: z = 1 + 3 + 1, xs >= 0, z' = 1 + (1 + 0 + (1 + 1 + x7)) + xs, x7 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, app(1, x12)), 5), 1 + 5 + x12), xs) :|: xs >= 0, x12 >= 0, z = 1 + 3 + 5, z' = 1 + (1 + 0 + (1 + 5 + x12)) + xs app(z, z') -{ 2 }-> app(app(app(app(4, 7), 1 + 6 + (z - 11)), 7), z' - 9) :|: z' - 9 >= 0, z - 11 >= 0 app(z, z') -{ 2 }-> app(app(app(app(4, 7), 1 + 3 + (z - 8)), 7), z' - 9) :|: z' - 9 >= 0, z - 8 >= 0 app(z, z') -{ 1 }-> app(app(app(app(4, 0), z - 4), x), xs) :|: z - 4 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(app(app(app(4, 1 + (z - 4) + x), z - 4), x), xs) :|: z - 4 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(app(app(4, app(z - 11, x5)), z - 11), x5), xs'')), app(0, xs)) :|: xs >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x5) + xs'')) + xs, x5 >= 0, xs'' >= 0, z - 11 >= 0 app(z, z') -{ 2 }-> app(app(0, app(app(app(app(4, app(z - 11, x5)), z - 11), x5), xs'')), app(1 + 6 + (1 + 3 + (z - 11)), xs)) :|: xs >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x5) + xs'')) + xs, x5 >= 0, xs'' >= 0, z - 11 >= 0 app(z, z') -{ 2 }-> app(app(0, app(app(3, fun1), x)), app(0, xs)) :|: xs >= 0, x4 >= 0, z = 1 + 6 + (1 + (1 + (1 + 4 + 2) + fun1) + x4), x >= 0, fun1 >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(3, fun1), x)), app(1 + 6 + (1 + (1 + (1 + 4 + 2) + fun1) + x4), xs)) :|: xs >= 0, x4 >= 0, z = 1 + 6 + (1 + (1 + (1 + 4 + 2) + fun1) + x4), x >= 0, fun1 >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(0, x3), app(app(3, fun''), x))), app(0, xs)) :|: xs >= 0, x >= 0, fun'' >= 0, x3 >= 0, z = 1 + 6 + (1 + (1 + (1 + 4 + 8) + fun'') + x3), z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(0, x3), app(app(3, fun''), x))), app(1 + 6 + (1 + (1 + (1 + 4 + 8) + fun'') + x3), xs)) :|: xs >= 0, x >= 0, fun'' >= 0, x3 >= 0, z = 1 + 6 + (1 + (1 + (1 + 4 + 8) + fun'') + x3), z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(app(0, app(app(0, app(z - 14, x2)), app(app(6, z - 14), xs'))), app(0, xs)) :|: xs >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x2) + xs')) + xs, xs' >= 0, z - 14 >= 0, x2 >= 0 app(z, z') -{ 2 }-> app(app(0, app(app(0, app(z - 14, x2)), app(app(6, z - 14), xs'))), app(1 + 6 + (1 + 6 + (z - 14)), xs)) :|: xs >= 0, z' = 1 + (1 + 0 + (1 + (1 + 0 + x2) + xs')) + xs, xs' >= 0, z - 14 >= 0, x2 >= 0 app(z, z') -{ 2 }-> app(app(0, app(5, app(1, x''))), app(0, xs)) :|: xs >= 0, z = 1 + 6 + 1, x'' >= 0, z' = 1 + (1 + 0 + (1 + 1 + x'')) + xs app(z, z') -{ 2 }-> app(app(0, app(5, app(1, x''))), app(1 + 6 + 1, xs)) :|: xs >= 0, z = 1 + 6 + 1, x'' >= 0, z' = 1 + (1 + 0 + (1 + 1 + x'')) + xs app(z, z') -{ 2 }-> app(app(0, app(1, x6)), app(0, xs)) :|: z' = 1 + (1 + 0 + (1 + 5 + x6)) + xs, xs >= 0, z = 1 + 6 + 5, x6 >= 0 app(z, z') -{ 2 }-> app(app(0, app(1, x6)), app(1 + 6 + 5, xs)) :|: z' = 1 + (1 + 0 + (1 + 5 + x6)) + xs, xs >= 0, z = 1 + 6 + 5, x6 >= 0 app(z, z') -{ 2 }-> app(app(0, 7), app(0, z' - 9)) :|: z - 14 >= 0, z' - 9 >= 0 app(z, z') -{ 2 }-> app(app(0, 7), app(0, z' - 9)) :|: z' - 9 >= 0, z - 11 >= 0 app(z, z') -{ 2 }-> app(app(0, 7), app(1 + 6 + (1 + 6 + (z - 14)), z' - 9)) :|: z - 14 >= 0, z' - 9 >= 0 app(z, z') -{ 2 }-> app(app(0, 7), app(1 + 6 + (1 + 3 + (z - 11)), z' - 9)) :|: z' - 9 >= 0, z - 11 >= 0 app(z, z') -{ 1 }-> app(app(0, 0), app(0, xs)) :|: z - 7 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(app(0, 0), app(1 + 6 + (z - 7), xs)) :|: z - 7 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(app(0, 1 + (z - 7) + x), app(0, xs)) :|: z - 7 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(app(0, 1 + (z - 7) + x), app(1 + 6 + (z - 7), xs)) :|: z - 7 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 2 }-> app(5, app(5, app(1, z' - 4))) :|: z = 1, z' - 4 >= 0 app(z, z') -{ 1 }-> app(5, 0) :|: z = 1, z' - 2 >= 0 app(z, z') -{ 1 }-> app(5, 1 + 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0 app(z, z') -{ 1 }-> app(1, z' - 6) :|: z' - 6 >= 0, z = 5 app(z, z') -{ 1 }-> app(0, z') :|: z = 1 + (1 + (1 + 4 + 2) + fun) + x, fun >= 0, z' >= 0, x >= 0 app(z, z') -{ 1 }-> app(0, app(0, z')) :|: fun >= 0, z' >= 0, z = 1 + (1 + (1 + 4 + 8) + fun) + x, x >= 0 app(z, z') -{ 1 }-> app(0, app(1 + 3 + fun, z')) :|: fun >= 0, z' >= 0, z = 1 + (1 + (1 + 4 + 8) + fun) + x, x >= 0 app(z, z') -{ 1 }-> app(1 + 3 + fun, z') :|: z = 1 + (1 + (1 + 4 + 2) + fun) + x, fun >= 0, z' >= 0, x >= 0 app(z, z') -{ 1 }-> app(1 + 0 + x, app(0, z')) :|: fun >= 0, z' >= 0, z = 1 + (1 + (1 + 4 + 8) + fun) + x, x >= 0 app(z, z') -{ 1 }-> app(1 + 0 + x, app(1 + 3 + fun, z')) :|: fun >= 0, z' >= 0, z = 1 + (1 + (1 + 4 + 8) + fun) + x, x >= 0 app(z, z') -{ 1 }-> 7 :|: z - 7 >= 0, z' = 7 app(z, z') -{ 1 }-> 7 :|: z - 4 >= 0, z' = 7 app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 app(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 8 :|: z = 8 encArg(z) -{ 0 }-> 7 :|: z = 7 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_app(z, z') -{ 0 }-> app(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_cons -{ 0 }-> 0 :|: encode_f -{ 0 }-> 1 :|: encode_f -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 3 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 4 :|: encode_filter2 -{ 0 }-> 0 :|: encode_g -{ 0 }-> 5 :|: encode_g -{ 0 }-> 0 :|: encode_map -{ 0 }-> 6 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 7 :|: encode_nil -{ 0 }-> 0 :|: encode_true -{ 0 }-> 8 :|: encode_true -{ 0 }-> 0 :|: ---------------------------------------- (143) 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_f -> null_encode_f [0] encode_g -> null_encode_g [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_f, null_encode_g, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app ---------------------------------------- (144) 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(f, c_app(f, x)) -> app(g, app(f, x)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(g, c_app(g, x)) -> app(f, x) [1] encArg(f) -> f [0] encArg(g) -> g [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_f -> f [0] encode_g -> g [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_f -> null_encode_f [0] encode_g -> null_encode_g [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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app g :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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_f :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 :: f:c_app:g:map:cons:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_g: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 ---------------------------------------- (145) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: f => 1 g => 5 map => 6 cons => 0 filter2 => 4 true => 8 filter => 3 false => 2 nil => 7 null_encArg => 0 null_encode_app => 0 null_encode_f => 0 null_encode_g => 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 ---------------------------------------- (146) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> app(app(app(app(4, app(fun, x)), fun), x), xs) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 3 + fun, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(app(3, fun), xs) :|: z = 1 + (1 + (1 + 4 + 2) + fun) + x, fun >= 0, xs >= 0, x >= 0, z' = xs app(z, z') -{ 1 }-> app(app(0, x), app(app(3, fun), xs)) :|: fun >= 0, xs >= 0, z = 1 + (1 + (1 + 4 + 8) + fun) + x, x >= 0, z' = xs app(z, z') -{ 1 }-> app(app(0, app(fun, x)), app(app(6, fun), xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 6 + fun, z' = 1 + (1 + 0 + x) + xs app(z, z') -{ 1 }-> app(5, app(1, x)) :|: z = 1, z' = 1 + 1 + x, x >= 0 app(z, z') -{ 1 }-> app(1, x) :|: z' = 1 + 5 + x, x >= 0, z = 5 app(z, z') -{ 1 }-> 7 :|: fun >= 0, z' = 7, z = 1 + 6 + fun app(z, z') -{ 1 }-> 7 :|: fun >= 0, z' = 7, z = 1 + 3 + fun app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 8 :|: z = 8 encArg(z) -{ 0 }-> 7 :|: z = 7 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_cons -{ 0 }-> 0 :|: encode_f -{ 0 }-> 1 :|: encode_f -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 3 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 4 :|: encode_filter2 -{ 0 }-> 0 :|: encode_g -{ 0 }-> 5 :|: encode_g -{ 0 }-> 0 :|: encode_map -{ 0 }-> 6 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 7 :|: encode_nil -{ 0 }-> 0 :|: encode_true -{ 0 }-> 8 :|: encode_true -{ 0 }-> 0 :|: Only complete derivations are relevant for the runtime complexity.