/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 189 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 6 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 14.3 s] (12) BOUNDS(1, INF) (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (14) TRS for Loop Detection (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) 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), 2 ms] (24) CdtProblem (25) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (26) CdtProblem (27) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 13 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 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), 0 ms] (42) CdtProblem (43) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 58 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 21 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), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 47 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (110) CdtProblem (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (118) CdtProblem (119) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (122) CpxRelTRS (123) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CpxWeightedTrs (125) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CpxTypedWeightedTrs (127) CompletionProof [UPPER BOUND(ID), 0 ms] (128) CpxTypedWeightedCompleteTrs (129) NarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (130) CpxTypedWeightedCompleteTrs (131) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (132) CpxRNTS (133) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CpxRNTS (135) CompletionProof [UPPER BOUND(ID), 0 ms] (136) CpxTypedWeightedCompleteTrs (137) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms] (138) 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(g, app(h, app(g, x))) -> app(g, x) app(g, app(g, x)) -> app(g, app(h, app(g, x))) app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), 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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f 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(g, app(h, app(g, x))) -> app(g, x) app(g, app(g, x)) -> app(g, app(h, app(g, x))) app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), 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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f 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(g, app(h, app(g, x))) -> app(g, x) app(g, app(g, x)) -> app(g, app(h, app(g, x))) app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), 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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(g, app(h, app(g, x))) -> app(g, x) app(g, app(g, x)) -> app(g, app(h, app(g, x))) app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), 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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: app(g, app(h, app(g, x))) -> app(g, x) app(g, app(g, x)) -> app(g, app(h, app(g, x))) app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), 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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app g :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app h :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app f :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app map :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app nil :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app cons :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app filter :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app true :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app false :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encArg :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_g :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_h :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_f :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app hole_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app1_0 :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app gen_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app2_0 :: Nat -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: app(g, app(h, app(g, x))) -> app(g, x) app(g, app(g, x)) -> app(g, app(h, app(g, x))) app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), 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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app g :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app h :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app f :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app map :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app nil :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app cons :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app filter :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app true :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app false :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encArg :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_g :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_h :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_f :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app hole_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app1_0 :: g:h:f:map:nil:cons:filter:filter2:true:false:cons_app gen_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app2_0 :: Nat -> g:h:f:map:nil:cons:filter:filter2:true:false:cons_app Generator Equations: gen_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app2_0(0) <=> g gen_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app2_0(+(x, 1)) <=> cons_app(g, gen_g:h:f: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 ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app2_0(n5219_0)) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app2_0(0)) Induction Step: encArg(gen_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app2_0(+(n5219_0, 1))) ->_R^Omega(0) app(encArg(g), encArg(gen_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app2_0(n5219_0))) ->_R^Omega(0) app(g, encArg(gen_g:h:f:map:nil:cons:filter:filter2:true:false:cons_app2_0(n5219_0))) ->_IH app(g, *3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (12) BOUNDS(1, INF) ---------------------------------------- (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(g, app(h, app(g, x))) -> app(g, x) app(g, app(g, x)) -> app(g, app(h, app(g, x))) app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), 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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(g, app(h, app(g, x))) -> app(g, x) app(g, app(g, x)) -> app(g, app(h, app(g, x))) app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), 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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f 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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(g) -> c ENCARG(h) -> c1 ENCARG(f) -> c2 ENCARG(map) -> c3 ENCARG(nil) -> c4 ENCARG(cons) -> c5 ENCARG(filter) -> c6 ENCARG(filter2) -> c7 ENCARG(true) -> c8 ENCARG(false) -> c9 ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_G -> c12 ENCODE_H -> c13 ENCODE_F -> c14 ENCODE_MAP -> c15 ENCODE_NIL -> c16 ENCODE_CONS -> c17 ENCODE_FILTER -> c18 ENCODE_FILTER2 -> c19 ENCODE_TRUE -> c20 ENCODE_FALSE -> c21 APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(h, app(g, z0)), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, app(app(f, app(h, z0)), z0)), APP(app(f, app(h, z0)), z0), APP(f, app(h, z0)), APP(h, z0)) APP(app(map, z0), nil) -> c25 APP(app(map, z0), app(app(cons, z1), z2)) -> c26(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) -> c27 APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(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) -> c29(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) -> c30(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(h, app(g, z0)), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, app(app(f, app(h, z0)), z0)), APP(app(f, app(h, z0)), z0), APP(f, app(h, z0)), APP(h, z0)) APP(app(map, z0), nil) -> c25 APP(app(map, z0), app(app(cons, z1), z2)) -> c26(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) -> c27 APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(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) -> c29(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) -> c30(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_g, encode_h, encode_f, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, ENCODE_G, ENCODE_H, ENCODE_F, ENCODE_MAP, ENCODE_NIL, ENCODE_CONS, ENCODE_FILTER, ENCODE_FILTER2, ENCODE_TRUE, ENCODE_FALSE, APP_2 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10_3, c11_3, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22_1, c23_3, c24_4, c25, c26_5, c27, c28_5, c29_4, c30_2 ---------------------------------------- (19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 20 trailing nodes: ENCODE_FILTER -> c18 ENCARG(filter) -> c6 ENCARG(nil) -> c4 ENCARG(cons) -> c5 ENCODE_NIL -> c16 ENCARG(map) -> c3 ENCODE_MAP -> c15 ENCARG(h) -> c1 ENCARG(false) -> c9 ENCARG(f) -> c2 ENCARG(g) -> c ENCODE_CONS -> c17 ENCODE_FALSE -> c21 ENCODE_FILTER2 -> c19 ENCODE_H -> c13 ENCODE_TRUE -> c20 ENCODE_G -> c12 ENCODE_F -> c14 ENCARG(true) -> c8 ENCARG(filter2) -> c7 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(h, app(g, z0)), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, app(app(f, app(h, z0)), z0)), APP(app(f, app(h, z0)), z0), APP(f, app(h, z0)), APP(h, z0)) APP(app(map, z0), nil) -> c25 APP(app(map, z0), app(app(cons, z1), z2)) -> c26(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) -> c27 APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(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) -> c29(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) -> c30(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(h, app(g, z0)), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, app(app(f, app(h, z0)), z0)), APP(app(f, app(h, z0)), z0), APP(f, app(h, z0)), APP(h, z0)) APP(app(map, z0), nil) -> c25 APP(app(map, z0), app(app(cons, z1), z2)) -> c26(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) -> c27 APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(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) -> c29(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) -> c30(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_g, encode_h, encode_f, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c10_3, c11_3, c22_1, c23_3, c24_4, c25, c26_5, c27, c28_5, c29_4, c30_2 ---------------------------------------- (21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 14 trailing tuple parts ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_g, encode_h, encode_f, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c10_3, c11_3, c22_1, c25, c27, c23_2, c24_1, c26_2, c28_2, c29_1, c30_1 ---------------------------------------- (23) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_g, encode_h, encode_f, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c10_3, c22_1, c25, c27, c23_2, c24_1, c26_2, c28_2, c29_1, c30_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(g) -> g encArg(h) -> h encArg(f) -> f 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_g -> g encode_h -> h encode_f -> f encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_g, encode_h, encode_f, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c10_3, c22_1, c25, c27, c23_2, c24_1, c26_2, c28_2, c29_1, c30_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_g -> g encode_h -> h encode_f -> f 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(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c10_3, c22_1, c25, c27, c23_2, c24_1, c26_2, c28_2, c29_1, c30_1, c_1 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0), ENCARG(g)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0), ENCARG(h)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0), ENCARG(f)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(g), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(h), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(APP(f, encArg(x1)), ENCARG(f), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0), ENCARG(g)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0), ENCARG(h)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0), ENCARG(f)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(g), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(h), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(APP(f, encArg(x1)), ENCARG(f), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c28_2, c29_1, c30_1, c_1, c10_3 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 28 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c28_2, c29_1, c30_1, c_1, c10_3, c10_2, c10_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) by APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(app(app(app(filter2, app(g, z0)), g), app(h, app(g, z0))), x2), APP(g, app(h, app(g, z0)))) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, z0)))), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(app(app(app(filter2, app(h, app(app(f, app(h, z0)), z0))), h), app(h, z0)), x2), APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(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)) -> c28(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)) -> c28(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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(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)) -> c28(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)) -> c28(APP(app(app(app(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(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(app(app(app(filter2, app(g, z0)), g), app(h, app(g, z0))), x2), APP(g, app(h, app(g, z0)))) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, z0)))), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(app(app(app(filter2, app(h, app(app(f, app(h, z0)), z0))), h), app(h, z0)), x2), APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(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)) -> c28(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)) -> c28(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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(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)) -> c28(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)) -> c28(APP(app(app(app(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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(app(app(app(filter2, app(g, z0)), g), app(h, app(g, z0))), x2), APP(g, app(h, app(g, z0)))) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, z0)))), g), app(g, z0)), x2), APP(g, app(g, z0))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(app(app(app(filter2, app(h, app(app(f, app(h, z0)), z0))), h), app(h, z0)), x2), APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(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)) -> c28(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)) -> c28(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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(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)) -> c28(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)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c_1, c10_3, c10_2, c10_1, c28_2 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c_1, c10_3, c10_2, c10_1, c28_2, c28_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, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, encArg(x1))) ENCODE_APP(f, x1) -> c(APP(f, 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(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, encArg(x1))) ENCODE_APP(f, x1) -> c(APP(f, 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_3, c10_2, c10_1, c28_2, c28_1, c_1 ---------------------------------------- (39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing nodes: ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_3, c10_2, c10_1, c28_2, c28_1, c_1 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by ENCARG(cons_app(x0, cons_app(x1, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(g), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(h), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(APP(f, app(encArg(x1), encArg(x2))), ENCARG(f), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(g), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(h), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(APP(f, app(encArg(x1), encArg(x2))), ENCARG(f), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_3, c10_2, c10_1, c28_2, c28_1, c_1 ---------------------------------------- (43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 18 trailing tuple parts ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_3, c10_2, c10_1, c28_2, c28_1, c_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1)), ENCARG(g)) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1)), ENCARG(h)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1)), ENCARG(f)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, x1), cons_app(z0, z1))) -> c10(APP(app(encArg(x0), encArg(x1)), app(encArg(z0), encArg(z1))), ENCARG(cons_app(x0, x1)), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1)), ENCARG(g)) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1)), ENCARG(h)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1)), ENCARG(f)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, g)) -> c10(APP(encArg(x0), g), ENCARG(x0)) by ENCARG(cons_app(g, g)) -> c10(APP(g, g), ENCARG(g)) ENCARG(cons_app(h, g)) -> c10(APP(h, g), ENCARG(h)) ENCARG(cons_app(f, g)) -> c10(APP(f, g), ENCARG(f)) ENCARG(cons_app(map, g)) -> c10(APP(map, g), ENCARG(map)) ENCARG(cons_app(nil, g)) -> c10(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(cons, g)) -> c10(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(filter, g)) -> c10(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(filter2, g)) -> c10(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(true, g)) -> c10(APP(true, g), ENCARG(true)) ENCARG(cons_app(false, g)) -> c10(APP(false, g), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), g)) -> c10(APP(app(encArg(z0), encArg(z1)), g), ENCARG(cons_app(z0, z1))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, g)) -> c10(APP(g, g), ENCARG(g)) ENCARG(cons_app(h, g)) -> c10(APP(h, g), ENCARG(h)) ENCARG(cons_app(f, g)) -> c10(APP(f, g), ENCARG(f)) ENCARG(cons_app(map, g)) -> c10(APP(map, g), ENCARG(map)) ENCARG(cons_app(nil, g)) -> c10(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(cons, g)) -> c10(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(filter, g)) -> c10(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(filter2, g)) -> c10(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(true, g)) -> c10(APP(true, g), ENCARG(true)) ENCARG(cons_app(false, g)) -> c10(APP(false, g), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(filter2, g)) -> c10(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(map, g)) -> c10(APP(map, g), ENCARG(map)) ENCARG(cons_app(nil, g)) -> c10(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(g, g)) -> c10(APP(g, g), ENCARG(g)) ENCARG(cons_app(h, g)) -> c10(APP(h, g), ENCARG(h)) ENCARG(cons_app(filter, g)) -> c10(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(false, g)) -> c10(APP(false, g), ENCARG(false)) ENCARG(cons_app(f, g)) -> c10(APP(f, g), ENCARG(f)) ENCARG(cons_app(true, g)) -> c10(APP(true, g), ENCARG(true)) ENCARG(cons_app(cons, g)) -> c10(APP(cons, g), ENCARG(cons)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, h)) -> c10(APP(encArg(x0), h), ENCARG(x0)) by ENCARG(cons_app(g, h)) -> c10(APP(g, h), ENCARG(g)) ENCARG(cons_app(h, h)) -> c10(APP(h, h), ENCARG(h)) ENCARG(cons_app(f, h)) -> c10(APP(f, h), ENCARG(f)) ENCARG(cons_app(map, h)) -> c10(APP(map, h), ENCARG(map)) ENCARG(cons_app(nil, h)) -> c10(APP(nil, h), ENCARG(nil)) ENCARG(cons_app(cons, h)) -> c10(APP(cons, h), ENCARG(cons)) ENCARG(cons_app(filter, h)) -> c10(APP(filter, h), ENCARG(filter)) ENCARG(cons_app(filter2, h)) -> c10(APP(filter2, h), ENCARG(filter2)) ENCARG(cons_app(true, h)) -> c10(APP(true, h), ENCARG(true)) ENCARG(cons_app(false, h)) -> c10(APP(false, h), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), h)) -> c10(APP(app(encArg(z0), encArg(z1)), h), ENCARG(cons_app(z0, z1))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, h)) -> c10(APP(g, h), ENCARG(g)) ENCARG(cons_app(h, h)) -> c10(APP(h, h), ENCARG(h)) ENCARG(cons_app(f, h)) -> c10(APP(f, h), ENCARG(f)) ENCARG(cons_app(map, h)) -> c10(APP(map, h), ENCARG(map)) ENCARG(cons_app(nil, h)) -> c10(APP(nil, h), ENCARG(nil)) ENCARG(cons_app(cons, h)) -> c10(APP(cons, h), ENCARG(cons)) ENCARG(cons_app(filter, h)) -> c10(APP(filter, h), ENCARG(filter)) ENCARG(cons_app(filter2, h)) -> c10(APP(filter2, h), ENCARG(filter2)) ENCARG(cons_app(true, h)) -> c10(APP(true, h), ENCARG(true)) ENCARG(cons_app(false, h)) -> c10(APP(false, h), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(true, h)) -> c10(APP(true, h), ENCARG(true)) ENCARG(cons_app(false, h)) -> c10(APP(false, h), ENCARG(false)) ENCARG(cons_app(filter2, h)) -> c10(APP(filter2, h), ENCARG(filter2)) ENCARG(cons_app(map, h)) -> c10(APP(map, h), ENCARG(map)) ENCARG(cons_app(cons, h)) -> c10(APP(cons, h), ENCARG(cons)) ENCARG(cons_app(h, h)) -> c10(APP(h, h), ENCARG(h)) ENCARG(cons_app(f, h)) -> c10(APP(f, h), ENCARG(f)) ENCARG(cons_app(filter, h)) -> c10(APP(filter, h), ENCARG(filter)) ENCARG(cons_app(nil, h)) -> c10(APP(nil, h), ENCARG(nil)) ENCARG(cons_app(g, h)) -> c10(APP(g, h), ENCARG(g)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, f)) -> c10(APP(encArg(x0), f), ENCARG(x0)) by ENCARG(cons_app(g, f)) -> c10(APP(g, f), ENCARG(g)) ENCARG(cons_app(h, f)) -> c10(APP(h, f), ENCARG(h)) ENCARG(cons_app(f, f)) -> c10(APP(f, f), ENCARG(f)) ENCARG(cons_app(map, f)) -> c10(APP(map, f), ENCARG(map)) ENCARG(cons_app(nil, f)) -> c10(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(cons, f)) -> c10(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(filter, f)) -> c10(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(filter2, f)) -> c10(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(true, f)) -> c10(APP(true, f), ENCARG(true)) ENCARG(cons_app(false, f)) -> c10(APP(false, f), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), f)) -> c10(APP(app(encArg(z0), encArg(z1)), f), ENCARG(cons_app(z0, z1))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, f)) -> c10(APP(g, f), ENCARG(g)) ENCARG(cons_app(h, f)) -> c10(APP(h, f), ENCARG(h)) ENCARG(cons_app(f, f)) -> c10(APP(f, f), ENCARG(f)) ENCARG(cons_app(map, f)) -> c10(APP(map, f), ENCARG(map)) ENCARG(cons_app(nil, f)) -> c10(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(cons, f)) -> c10(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(filter, f)) -> c10(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(filter2, f)) -> c10(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(true, f)) -> c10(APP(true, f), ENCARG(true)) ENCARG(cons_app(false, f)) -> c10(APP(false, f), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(filter, f)) -> c10(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(cons, f)) -> c10(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(h, f)) -> c10(APP(h, f), ENCARG(h)) ENCARG(cons_app(filter2, f)) -> c10(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(map, f)) -> c10(APP(map, f), ENCARG(map)) ENCARG(cons_app(g, f)) -> c10(APP(g, f), ENCARG(g)) ENCARG(cons_app(false, f)) -> c10(APP(false, f), ENCARG(false)) ENCARG(cons_app(true, f)) -> c10(APP(true, f), ENCARG(true)) ENCARG(cons_app(f, f)) -> c10(APP(f, f), ENCARG(f)) ENCARG(cons_app(nil, f)) -> c10(APP(nil, f), ENCARG(nil)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) by ENCARG(cons_app(g, map)) -> c10(APP(g, map), ENCARG(g)) ENCARG(cons_app(h, map)) -> c10(APP(h, map), ENCARG(h)) ENCARG(cons_app(f, map)) -> c10(APP(f, map), ENCARG(f)) ENCARG(cons_app(map, map)) -> c10(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c10(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c10(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c10(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c10(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c10(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c10(APP(false, map), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), map)) -> c10(APP(app(encArg(z0), encArg(z1)), map), ENCARG(cons_app(z0, z1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, map)) -> c10(APP(g, map), ENCARG(g)) ENCARG(cons_app(h, map)) -> c10(APP(h, map), ENCARG(h)) ENCARG(cons_app(f, map)) -> c10(APP(f, map), ENCARG(f)) ENCARG(cons_app(map, map)) -> c10(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c10(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c10(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c10(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c10(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c10(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c10(APP(false, map), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(filter2, map)) -> c10(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(map, map)) -> c10(APP(map, map), ENCARG(map)) ENCARG(cons_app(g, map)) -> c10(APP(g, map), ENCARG(g)) ENCARG(cons_app(false, map)) -> c10(APP(false, map), ENCARG(false)) ENCARG(cons_app(filter, map)) -> c10(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(nil, map)) -> c10(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(h, map)) -> c10(APP(h, map), ENCARG(h)) ENCARG(cons_app(true, map)) -> c10(APP(true, map), ENCARG(true)) ENCARG(cons_app(f, map)) -> c10(APP(f, map), ENCARG(f)) ENCARG(cons_app(cons, map)) -> c10(APP(cons, map), ENCARG(cons)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) by ENCARG(cons_app(g, nil)) -> c10(APP(g, nil), ENCARG(g)) ENCARG(cons_app(h, nil)) -> c10(APP(h, nil), ENCARG(h)) ENCARG(cons_app(f, nil)) -> c10(APP(f, nil), ENCARG(f)) ENCARG(cons_app(map, nil)) -> c10(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c10(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c10(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c10(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c10(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c10(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c10(APP(false, nil), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), nil)) -> c10(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, nil)) -> c10(APP(g, nil), ENCARG(g)) ENCARG(cons_app(h, nil)) -> c10(APP(h, nil), ENCARG(h)) ENCARG(cons_app(f, nil)) -> c10(APP(f, nil), ENCARG(f)) ENCARG(cons_app(map, nil)) -> c10(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c10(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c10(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c10(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c10(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c10(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c10(APP(false, nil), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(h, nil)) -> c10(APP(h, nil), ENCARG(h)) ENCARG(cons_app(f, nil)) -> c10(APP(f, nil), ENCARG(f)) ENCARG(cons_app(true, nil)) -> c10(APP(true, nil), ENCARG(true)) ENCARG(cons_app(filter2, nil)) -> c10(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(map, nil)) -> c10(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c10(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(filter, nil)) -> c10(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(g, nil)) -> c10(APP(g, nil), ENCARG(g)) ENCARG(cons_app(cons, nil)) -> c10(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(false, nil)) -> c10(APP(false, nil), ENCARG(false)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) by ENCARG(cons_app(g, cons)) -> c10(APP(g, cons), ENCARG(g)) ENCARG(cons_app(h, cons)) -> c10(APP(h, cons), ENCARG(h)) ENCARG(cons_app(f, cons)) -> c10(APP(f, cons), ENCARG(f)) ENCARG(cons_app(map, cons)) -> c10(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c10(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c10(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c10(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c10(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c10(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c10(APP(false, cons), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), cons)) -> c10(APP(app(encArg(z0), encArg(z1)), cons), ENCARG(cons_app(z0, z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, cons)) -> c10(APP(g, cons), ENCARG(g)) ENCARG(cons_app(h, cons)) -> c10(APP(h, cons), ENCARG(h)) ENCARG(cons_app(f, cons)) -> c10(APP(f, cons), ENCARG(f)) ENCARG(cons_app(map, cons)) -> c10(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c10(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c10(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c10(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c10(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c10(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c10(APP(false, cons), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(false, cons)) -> c10(APP(false, cons), ENCARG(false)) ENCARG(cons_app(filter2, cons)) -> c10(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(map, cons)) -> c10(APP(map, cons), ENCARG(map)) ENCARG(cons_app(true, cons)) -> c10(APP(true, cons), ENCARG(true)) ENCARG(cons_app(filter, cons)) -> c10(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(g, cons)) -> c10(APP(g, cons), ENCARG(g)) ENCARG(cons_app(h, cons)) -> c10(APP(h, cons), ENCARG(h)) ENCARG(cons_app(cons, cons)) -> c10(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(nil, cons)) -> c10(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(f, cons)) -> c10(APP(f, cons), ENCARG(f)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) by ENCARG(cons_app(g, filter)) -> c10(APP(g, filter), ENCARG(g)) ENCARG(cons_app(h, filter)) -> c10(APP(h, filter), ENCARG(h)) ENCARG(cons_app(f, filter)) -> c10(APP(f, filter), ENCARG(f)) ENCARG(cons_app(map, filter)) -> c10(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c10(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c10(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c10(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c10(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c10(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c10(APP(false, filter), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter)) -> c10(APP(app(encArg(z0), encArg(z1)), filter), ENCARG(cons_app(z0, z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, filter)) -> c10(APP(g, filter), ENCARG(g)) ENCARG(cons_app(h, filter)) -> c10(APP(h, filter), ENCARG(h)) ENCARG(cons_app(f, filter)) -> c10(APP(f, filter), ENCARG(f)) ENCARG(cons_app(map, filter)) -> c10(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c10(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c10(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c10(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c10(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c10(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c10(APP(false, filter), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(f, filter)) -> c10(APP(f, filter), ENCARG(f)) ENCARG(cons_app(nil, filter)) -> c10(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(true, filter)) -> c10(APP(true, filter), ENCARG(true)) ENCARG(cons_app(cons, filter)) -> c10(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter2, filter)) -> c10(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(map, filter)) -> c10(APP(map, filter), ENCARG(map)) ENCARG(cons_app(g, filter)) -> c10(APP(g, filter), ENCARG(g)) ENCARG(cons_app(false, filter)) -> c10(APP(false, filter), ENCARG(false)) ENCARG(cons_app(filter, filter)) -> c10(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(h, filter)) -> c10(APP(h, filter), ENCARG(h)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) by ENCARG(cons_app(g, filter2)) -> c10(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(h, filter2)) -> c10(APP(h, filter2), ENCARG(h)) ENCARG(cons_app(f, filter2)) -> c10(APP(f, filter2), ENCARG(f)) ENCARG(cons_app(map, filter2)) -> c10(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c10(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c10(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c10(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c10(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c10(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c10(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter2)) -> c10(APP(app(encArg(z0), encArg(z1)), filter2), ENCARG(cons_app(z0, z1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, filter2)) -> c10(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(h, filter2)) -> c10(APP(h, filter2), ENCARG(h)) ENCARG(cons_app(f, filter2)) -> c10(APP(f, filter2), ENCARG(f)) ENCARG(cons_app(map, filter2)) -> c10(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c10(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c10(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c10(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c10(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c10(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c10(APP(false, filter2), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(cons, filter2)) -> c10(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(h, filter2)) -> c10(APP(h, filter2), ENCARG(h)) ENCARG(cons_app(true, filter2)) -> c10(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(filter2, filter2)) -> c10(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(map, filter2)) -> c10(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(g, filter2)) -> c10(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(nil, filter2)) -> c10(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(false, filter2)) -> c10(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(f, filter2)) -> c10(APP(f, filter2), ENCARG(f)) ENCARG(cons_app(filter, filter2)) -> c10(APP(filter, filter2), ENCARG(filter)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) by ENCARG(cons_app(g, true)) -> c10(APP(g, true), ENCARG(g)) ENCARG(cons_app(h, true)) -> c10(APP(h, true), ENCARG(h)) ENCARG(cons_app(f, true)) -> c10(APP(f, true), ENCARG(f)) ENCARG(cons_app(map, true)) -> c10(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c10(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c10(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c10(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c10(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c10(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c10(APP(false, true), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), true)) -> c10(APP(app(encArg(z0), encArg(z1)), true), ENCARG(cons_app(z0, z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, true)) -> c10(APP(g, true), ENCARG(g)) ENCARG(cons_app(h, true)) -> c10(APP(h, true), ENCARG(h)) ENCARG(cons_app(f, true)) -> c10(APP(f, true), ENCARG(f)) ENCARG(cons_app(map, true)) -> c10(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c10(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c10(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c10(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c10(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c10(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c10(APP(false, true), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(false, true)) -> c10(APP(false, true), ENCARG(false)) ENCARG(cons_app(f, true)) -> c10(APP(f, true), ENCARG(f)) ENCARG(cons_app(filter, true)) -> c10(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(true, true)) -> c10(APP(true, true), ENCARG(true)) ENCARG(cons_app(nil, true)) -> c10(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(g, true)) -> c10(APP(g, true), ENCARG(g)) ENCARG(cons_app(filter2, true)) -> c10(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(map, true)) -> c10(APP(map, true), ENCARG(map)) ENCARG(cons_app(cons, true)) -> c10(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(h, true)) -> c10(APP(h, true), ENCARG(h)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) by ENCARG(cons_app(g, false)) -> c10(APP(g, false), ENCARG(g)) ENCARG(cons_app(h, false)) -> c10(APP(h, false), ENCARG(h)) ENCARG(cons_app(f, false)) -> c10(APP(f, false), ENCARG(f)) ENCARG(cons_app(map, false)) -> c10(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c10(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c10(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c10(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c10(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c10(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c10(APP(false, false), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), false)) -> c10(APP(app(encArg(z0), encArg(z1)), false), ENCARG(cons_app(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c10(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(h, cons_app(x1, x2))) -> c10(APP(h, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, g), x2)) -> c10(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, h), x2)) -> c10(APP(app(encArg(x0), h), encArg(x2)), ENCARG(cons_app(x0, h)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c10(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c10(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(h, x1), x2)) -> c10(APP(app(h, encArg(x1)), encArg(x2)), ENCARG(cons_app(h, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c10(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), h)) -> c10(APP(app(encArg(x0), encArg(x1)), h), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c10(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c10(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(g, false)) -> c10(APP(g, false), ENCARG(g)) ENCARG(cons_app(h, false)) -> c10(APP(h, false), ENCARG(h)) ENCARG(cons_app(f, false)) -> c10(APP(f, false), ENCARG(f)) ENCARG(cons_app(map, false)) -> c10(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c10(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c10(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c10(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c10(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c10(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c10(APP(false, false), ENCARG(false)) S tuples: APP(g, app(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22_1, c25, c27, c23_2, c24_1, c26_2, c29_1, c30_1, c10_2, c10_1, c28_2, c28_1, c_1, c10_3 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(false, false)) -> c10(APP(false, false), ENCARG(false)) ENCARG(cons_app(nil, false)) -> c10(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(f, false)) -> c10(APP(f, false), ENCARG(f)) ENCARG(cons_app(true, false)) -> c10(APP(true, false), ENCARG(true)) ENCARG(cons_app(cons, false)) -> c10(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(h, false)) -> c10(APP(h, false), ENCARG(h)) ENCARG(cons_app(g, false)) -> c10(APP(g, false), ENCARG(g)) ENCARG(cons_app(filter, false)) -> c10(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c10(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(map, false)) -> c10(APP(map, false), ENCARG(map)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(g) -> g encArg(h) -> h encArg(f) -> f 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(g, app(h, app(g, z0))) -> app(g, z0) app(g, app(g, z0)) -> app(g, app(h, app(g, z0))) app(h, app(h, z0)) -> app(h, app(app(f, app(h, z0)), 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(h, app(g, z0))) -> c22(APP(g, z0)) APP(app(map, z0), nil) -> c25 APP(app(filter, z0), nil) -> c27 APP(g, app(g, z0)) -> c23(APP(g, app(h, app(g, z0))), APP(g, z0)) APP(h, app(h, z0)) -> c24(APP(h, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) ENCARG(cons_app(g, x1)) -> c10(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(h, x1)) -> c10(APP(h, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1)) APP(app(filter, g), app(app(cons, app(g, z0)), x2)) -> c28(APP(app(app(app(filter2, app(g, app(h, app(g, 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)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, g), app(app(cons, app(h, app(g, z0))), x2)) -> c28(APP(g, app(h, app(g, z0)))) APP(app(filter, h), app(app(cons, app(h, z0)), x2)) -> c28(APP(h, app(h, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, h) -> c(APP(encArg(x0), h)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) 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(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(h, x1) -> c(APP(h, 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, g))) -> c10(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, h))) -> c10(APP(encArg(x0), app(encArg(x1), h)), ENCARG(x0), ENCARG(cons_app(x1, h))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c10(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c10(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(h, x2))) -> c10(APP(encArg(x0), app(h, encArg(x2))), ENCARG(x0), ENCARG(cons_app(h, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c10(APP(encArg(x0), app(f, encArg(x